H Høgaasen

P Sorba
email: sorba@lapp.in2p3.fr

On Quarks and Flavour Symmetry

Hadronic spectroscopy can be introduced to students and developed rather far without requiring SU(N) flavour symmetry. In such a "minimalist" presentation, we are naturally led to comment and clarify the concept of the "generalized" Pauli principle.

Introduction

Internal symmetry groups have a glorious place in the history of physics. One of the highlights was the discovery by Fermi that low energy pion-nucleon scattering is dominated by a single resonance of spin 3/2. By treating the nucleons as an isospin doublet , the pions as an isospin triplet and the resonance ∆ as a state with isospin 3/2, Fermi and his collaborators proved the dynamical fact that, to a good precision, isospin is conserved in strong interactions.

Another highlight was Gell-Mann's prediction of the spin 3/2 Ω -at the Geneva conference in 1962. The subsequent discovery of this state at Brookhaven in 1964 made everyone confident that the flavourgroup SU(3) was as relevant for strong interactions as the isospin group.

There is however a difference between the two cases we have mentioned:

Fermi's discovery was a discovery in the dynamics of particles while Gell-Mann's prediction is usually presented as coming from the assumption that baryons should fall into specific representations of SU(3)-flavour, namely the 8 dimensional one for the lightest spin 1/2 baryons, the 10 dimensional one for the lightest spin 3/2 baryons.

Experimental results from reactions involving the assignment of mesons and baryons to irreducible representations of SU(3) came later, and showed that the concept of "broken flavour symmetry" was useful.

Subsequently the flavour symmetry groups SU(4) and SU(5) have been invoked to classify the multiplicities of states when charm and bottom quantum numbers are added.

But let us stress that the assignment of a multiplet of particles to a group representation is in itself an empty statement. Its usefulness depends on the group to be an (almost) symmetry group. When Fermi's team discovered that the isospin group -historically, this group was first supposed to be R (3) before becoming SU(2) -it was by studying reactions. If a pion-nucleon state is a linear combination of isospin I = 1/2 and I = 3/2, only two independent amplitudes A 1/2 and A 3/2 will describe all πN → πN reactions 2 : this turned out to be correct.

Experiments must decide to which extent flavour symmetry is a useful concept. From the multiplication table of SU(3): 8 ⊗ 8 = 1 ⊕ 8 ⊕ 8 ⊕ 10 ⊕ 10 ⊕ 27 it follows that the 64 reactions we have in "quasi elastic" 2 ↔ 2 reactions between octet states are given by only six independent amplitudes. From the same table we see, as the representation 10 occurs only once, that in the coupling of a decuplet to two octets there is only one amplitude. This is similar to the decay of ∆ → N π in the isospin symmetric -SU(2) -case. So, if one has baryons made from u, d and s quarks, the assumption of flavour symmetry relates the coupling constant in ∆ ++ → P π + to the coupling constants in Σ * 0 → Λπ 0 , Σ * 0 → Σ + π -and Ξ * 0 → Ξ 0 π 0 . Experimentally they come out correctly within 10% .

1964 was the year Gell-Mann and Zweig invented quarks.

Right from the start Zweig was a fervent believer in the objective existence of quarks (that he called aces) as the fundamental constituents of hadrons.

With three flavours of aces (quarks) mesons came in qq nonets. There was no need to invoke a "magic mixing" between a flavour octet and a singlet; to explain that one of the vector mesons decayed into KK pairs was simply the reflection of the fact that it was composed of ss. If some "magic" is there , it is rather in the structure of the spin zero mesons.

In the years before quarks were generally accepted as physical beings in the physics community, flavour symmetry arguments were ( and often still are! ) regarded as more high-brow than arguments simply based on quarks.

But today hardly anybody denies that quarks exist. It is easy to criticize the (nonrelativistic) quark model where baryons are made from three valence quarks and mesons from a quark-antiquark pair, but it cannot be denied that it gives a set of rules to classify hadron states, and also to calculate many of their static properties which are quite illuminating.

However, as we will see in the following, we do not need any flavour SU(N) group either to classify hadrons or to estimate many of their physical properties. Hereafter, the fundamental symmetry groups we will use are the group of rotations in the three dimensional space represented by the SU (2) spin group ( together with the O(3) orbital angular momentum one in the case of excited states), and the colour SU(3) group.

Indeed, the SU(3) symmetry group of fundamental importance is not the group acting over flavours, but the group acting over colour space. It is a group that is gauged and thereby introduces gluonfields while defining QCD.

The (yet unproven) dogma that free particles are colourless, i.e. transform as a singlet under the colour SU(3) group, implies that the most economical configurations are those involving three quarks (baryon) or a quarkantiquark pair (meson). This comes from the decomposition of (colour) SU(3) representations:

3 ⊗ 3 ⊗ 3 = 1 ⊕ 8 ⊕ 8 ⊕ 10 and 3 ⊗ 3 = 6 ⊕ 3 3 ⊗ 3 = 1 ⊕ 8
where the representations 6 and 3 respectively involve symmetric and antisymmetric combinations of the two colour indices.

The second essential ingredient we need is the Pauli principle. Of course it is of importance only for baryons [START_REF] Bourbaki | Eléments de Mathématique[END_REF] , and it is actually this class of hadrons which deserves to be considered for our purpose. Indeed, at least with regard to their classification, qq mesons with a given spin and parity are simply gathered in multiplets of dimension N 2 , N being the number of different quark flavours.

Note that such three quark states are colour-antisymmetric under the exchange of two quarks. Then, the Pauli principle tells us that a state involving two identical quarks, i.e. quarks with identical flavours, must be symmetric in the other quantum numbers.

We will start by showing in Sections 2 and 3 that the multiplets of low-lying as well as excited baryonic resonances can be directly constructed with the above hypotheses and tools at hand. In Section 4, expressions for S-wave three quark states only satisfying permutational symmetry imposed by the Pauli principle are given and used to compute baryon mass splittings through (colour) magnetic interactions between the constituent quarks; their magnetic moments are also obtained.

However the basis constructed in Section 2 does not appear suitable for computations of physical quantities involving flavour changing forces, such as, for instance, semi-leptonic decays. A convenient way to circumvent this difficulty is to impose complete symmetry under permutations of the three constituents within the baryon wave function rather than the -partialsymmetry dictated by the Pauli principle: this prescription appeals to the so-called "generalized" Pauli principle, which, as is well known, does not introduce any extra physical assumption. The corresponding developments are presented in Section 5.

We conclude by commenting on the direct connection between irreducible representations of the unitary groups and their properties under the permutation group, which explains, in our opinion, their perfect adaptability to the classification of baryons.

In order not of overload the text and keep clear the main idea of this note, we have decided to treat in appendices the following points. Appendix A is devoted to a general and explicit construction of the (three state particle) Jacobi coordinates. In our knowledge, such coordinates have already been explicated in the case of the harmonic oscillator with only one coupling constant, but not with a different coupling associated to each couple of constituents.

Then we give a proof, in Appendix B, of the equivalence between the two Hilbert space descriptions introduced in Section 4 and 5 respectively, which involve states satisfying partial permutation symmetry (simple Pauli principle) for the former, and total permutation symmetry (generalized Pauli principle) for the later. The Fock space emerges rather naturally from this construction, and we show in Appendix C how it can be used to represent baryon states. Finally, we give in Appendix D the explicit wave-functions of the eight S = 1/2 low-lying baryons in the three bases successively in-troduced: the permutational "partial" symmetric basis (cf. Section 4), the totally symmetric basis ( cf. Section 5) and the Fock space basis, this last one combining the advantages of the other two representations: simplicity of the expressions and complete symmetry.

As this paper can be partly regarded as a set of lecture notes, there is an almost empty reference list. We apologize for this but we realize that to do justice to all the people who developed the subject, we would need a list longer than the article.

2 The ground state of three quarks This is the subject of most introductory courses on elementary particles. So, we will treat this first. Although the following is known, it is apparently not well known, as we realized by looking up at lectures notes on the web.

Among many popular monographs, we found only one [1] which shares the approach presented below.

We now forget about colour, remembering that each pair of quarks are antisymmetric in colour so that the baryon states must be symmetric in the other degrees of freedom. This was for a long time called the Dalitz symmetric quark model. Note at this point that it was realized [2] that if quarks were fermions such that there existed an hidden quantum number -which later turned out to be the colour -justifying the symmetric quark model, then the multiplicity of the ground states would follow independently of any flavour symmetry group.

In the ground state where no angular momentum is involved, the only degree freedom of each quark is the spin. As a pair of identical quarks must be symmetric under interchange it must be symmetric in spin: the spin of the pair is one.

Integrating out the spatial degrees of freedom, we are left with a Hamiltonian over the flavour-spin space of the quarks. 4 . Now let us count the ground states:

Suppose that we have N flavours (u, d, s, c,..) and we choose three of these to make baryons.

First for three identical flavours:

There are N ways of choosing these baryons made up of three identical quarks. Each pair has S=1 so all have total spin 3/2.

Two identical flavours:

There are N(N-1) ways of choosing two identical and one different from the two first chosen.

The identical quarks are coupled to spin one, and the total spin is S=3/2 or S=1/2.

All three quarks different:

There are N(N-1)(N-2)/6 ways of choosing three different flavours.The Pauli principle imposes no restriction here, so the total spin can be 3/2 or 1/2, the multiplicity of S=1/2 is two.

So the number N(3/2) of flavour states with total spin 3/2 is:

N(3/2) =N+N(N-1)+ N(N-1)(N-2)/6 = N(N+1)(N+2)/6
and the number of flavour states with spin 1/2 is:

N(1/2)= N(N-1)+2N(N-1)(N-2)/6 =N(N+1)(N-1)/3
We immediately see that we have found the same multiplicities as is commonly inferred from the dimensions of the representations of the flavour symmetry group SU(N).

For N=3 we have an "eightfold way", more precisely an octet of S=1/2 states and a decuplet of S=3/2 states.

But we also realize that these multiplicities have nothing to do with the existence of an internal symmetry group, they have their sole origin in the Pauli principle. Not only u,d,s, quarks, but also u,c,b, or any triplet of flavour quarks will provide with an octet and a decuplet. Now, let us count the total number of such quantum mechanical states.

Since there are 2S+1 states in a spin S representation, one gets:

N(total)= 4 N(N+1)(N+2)/6 +2 N(N+1)(N-1)/3 = 2 N(N+1)(2N+1)/3
and that is exactly the dimension of the SU(2N) completely symmetric representation arising from the tensorial product of three 2N dimensional fundamental SU(2N) representations. Taking as an example N= 3, we indeed get 56 states, that is the dimension of the corresponding symmetric representation of the usually called "flavour-spin" SU(6) group: we will comment later on this point, in direct connection with the "generalized" Pauli principle.

Excited states

Let us now turn our attention to the construction of P-wave baryon states, to which we will restrict our presentation in order not to overload this note.

These negative parity baryons are associated to an L=1 orbital momentum.

This unit of O(3) orbital angular momentum stands usually in one of the two quark relative coordinates, and the most common model which is well adapted to represent such an effect is that of the harmonic oscillator. The reason for treating confining forces between quarks using harmonic oscillator potentials is simply that the center of mass motion can be separated out using Jacobi coordinates. Note that this choice has no importance for our purpose, namely the number of exited states.

We will use as a Hamiltonian

H h.o. = i (m i + p 2 i /2m i ) + i<j (k/2) • ( -→ r i --→ r j ) 2 ) (1)
where -→ r i , i=1,2,3, denote the respective positions of the three quarks. Although we do not start with a more general Hamiltonian H gen , we can obtain the general solution for the spectrum by first using H h.o. to get states of a harmonic oscillator basis and then perturb them with the perturbation H gen -H h.o. . The energy of the levels will change with respect to the ones of the harmonic oscillator, but their number will be the same.

In terms of the Jacobi coordinates -→ ρ , -→ λ and the center of mass coordinates

--→ R cm , the Hamiltonian separates into H h.o. = i m i + H cm + H ρ + H λ with H cm = P 2 /2M, M = i m i , -→ P = i -→ p i , H ρ = p 2 ρ /2m + + 3k/2 • ( -→ ρ ) 2 (2)
H λ = p 2 λ /2m λ + 3k/2 • ( -→ λ ) 2 .
The explicit expressions of m + and m -are given in Appendix A, where a detailed construction of Jacobi coordinates is developed, and also generalized to the case where we have a different coupling k ij for each quark pair q i q j in the potential term.

For notational simplicity, we consider here the equal mass case, i.e. m i = m to which correspond: m + = m -= m and the relative coordinates -→ ρ and -→ λ reduce to :

-→ ρ = ( -→ r 1 --→ r 2 )/ √ 2 -→ λ = ( -→ r 1 + -→ r 2 -2 -→ r 3 )/ √ 6
So, let us separately study the different configurations:

i) Baryons made of three flavour identical quarks qqq:

Consider the oscillator relative to -→ ρ : it is antisymmetric in the exchange

(1) -( 2). The doublet made with the first two quarks qq , that is of quarks in position ( 1) and ( 2), has automatically spin S = 0, in order for the total spin and orbital momentum part to be symmetric ( Pauli principle ). It follows that in this -→ ρ configuration, the total spin of the baryon is S = 1/2, and the spin/orbit part of the wave function reads,for Sz = +1/2 and up to a normalization factor:

( -→ r 1 --→ r 2 ) (↑↓↑ -↓↑↑ ) + sym.
Replacing now -→ ρ by -→ λ , which is symmetric in the first two quarks, one easily deduces that the corresponding qq doublet has a (symmetric) spin S = 1. However in the product S = 1 by S = 1/2, the totally spin symmetric S = 3/2 part cannot provide, when combined with the (not completely symmetric) -→ λ , a completely symmetric wave function. It follows that the only possibility for the resulting baryon is to have spin 1/2. Moreover, one notes that the spin/orbit part of the wave function reads:

( -→ r 1 + -→ r 2 -2 -→ r 3 ) (2 ↑↑↓ -↓↑↑ -↑↓↑) + sym.
which is exactly the same, up to a scale factor, as the one obtained just above for the -→ ρ case.

Therefore, for qqq configurations, the only solution is given by the -→ ρ -with the -→ ρ oscillator: the qq spin part must be zero, leading for the (qqq') baryon to S = 1/2, with in the wave function a qq spin/orbit part proportional to:

( -→ r 1 --→ r 2 ) (↑↓ -↓↑ ) + sym.
-with the -→ λ oscillator: the qq spin part is now one, allowing the total spin to be S = 3/2 and S = 1/2.

iii) Baryons made of three different flavour quarks qq ′ q": Now there is no restriction imposed by the Pauli principle. It follows that with the -→ ρ as well as with the -→ λ oscillator, one can get one baryon with total spin S = 3/2 and two baryons of total spin S = 1/2 for each triplet of flavours.

As a conclusion, let us count the number of states of three different flavours:

-with S = 3/2: there are 6 states of qqq ′ -type with the -→ λ oscillator, one state qq ′ q" with each of the -→ ρ and -→ λ oscillator, that is a total of 8 states.

-with S = 1/2: there are 3 states of qqq-type made from the -→ ρ osc. In a similar manner, one can show that, for higher excitations, the number of excited states and their configurations are always what one should infer from the SU(6) approach.

Flavour nonchanging forces

Let us now focus on forces acting on the quarks (gluonic and electromagnetic in particular) that do not change the quark flavour.

We can represent any baryon made of the three quarks q 1 ,q 2 ,q 3 as follows:

B(q 1 , q 2 , q 3 ) = (q 1 q 2 ) s ⊗ (q 3 ) 1/2 ( 3
)
where s is the spin of the doublet of the quark pair q 1 q 2 . One obviously has S = 1 or S = 0.

If we denote by q ↑ i the i quark with spin up, by q ↓ i the corresponding spin down state, then with the help of Clebsch-Gordan coefficients, any S = 3/2 state with S z = 3/2 writes :

q ↑ 1 q ↑ 2 q ↑ 3 while it becomes for S z = 1/2 : [q ↑ 1 q ↑ 2 q ↓ 3 + q ↑ 1 q ↓ 2 q ↑ 3 + q ↓ 1 q ↑ 2 q ↑ 3 ]/ √ 3
and so on. As we have seen there are 10 such flavour states, if we limit ourselves to three flavours.

For the 8 flavour states with spin 1/2 and S z = +1/2, one must distinguish the case when the first two quarks are identical in flavour from the case when all flavours are different. In the first situation, the spin of q 1 and q 2 must couple to one ( we write below: q 1 = q 2 ) and Clebsch-Gordan coefficients give:

1 √ 6 [2 • q ↑ 1 q ↑ 1 q ↓ 3 -(q ↑ 1 q ↓ 1 + q ↓ 1 q ↑ 1 )q ↑ 3 ]
. In the second configuration, the Pauli principle gives no restriction, and we have two states, the spin of the two first quarks being coupled to produce either a spin one (ψ 1 ) or zero (ψ 2 ):

ψ 1 = 1 √ 6 [2 • q ↑ 1 q ↑ 2 q ↓ 3 -(q ↑ 1 q ↓ 2 + q ↓ 1 q ↑ 2 )q ↑ 3 ] ψ 2 = [(q ↑ 1 q ↓ 2 -q ↓ 1 q ↑ 2 )q ↑ 3 ]/ √ 2
It clearly does not matter which ordering we have for the different flavours, but it will be shown in a moment that it is often convenient to order them by placing the lightest quark(s) in front of the heaviest.

We stress again that the classification of states is into a spin 3/2 decuplet and a spin 1/2 octet, no matter which three flavours we choose. With quarks d, c and b we have the same structure as with the lightest states u, d and s.

From this remark it should be clear for students that the classification of states is one thing; the question whether we have a flavour symmetry group SU(3) is completely different. Now, from the 8 states with total spin 1/2 that we have constructed, we can span a Hilbert space and consider how interactions act on states therein.

If there were no spin dependent interquark forces, it is evident that the states made of the same quarks would have the same mass. The interquark Hamiltonian would be diagonal in our Hilbert space. This is definitely not so. In the beginning it seemed strange that the Λ and Σ 0 had different masses. An early explanation, based on spin-spin interactions, was given by Sakharov and Zeldowich in 1966 with the Hamiltonian :

H S = i,j c ij -→ σ i • -→ σ j (4)
all coefficients c ij being equal. In 1975 this spin-spin interaction was shown to be a consequence of QCD by De Rujula, Georgi and Glashow:

H CM = - i,j C ij λ i • λ j σ i . σ j (5)
As all quark pairs are in a 3 state of colour, the colour part λ i • λ j factorizes, giving a common factor -8/3, so that:

H CM = (8/3) i,j C ij σ i . σ j (6)
Here the coefficients C ij are, among other things, dependent on the quark masses and properties of the spatial wave functions of the quarks in the system.

A natural scaling assumption for the coefficients C ij ∝ 1 m i m j comes from the analogy with the hyperfine splitting in atoms that originates in the interaction between (electro) magnetic moments. In the physics of quarks the analog interaction is between the quarks colourmagnetic moments.

Let us now imagine that we have integrated out all the spatial variables for the three quarks. We can then write an effective Hamiltonian over the spin-space of the quark:

H = i m i + (8/3) i,j C ij σ i . σ j (7)
Here effective masses m i incorporate the masses of the quarks as well as their kinetic energy5 . If we assume that each effective mass m i is (almost) the same in different baryons, then we get mass formulae. For this, we have only to determine the eigenvalues of H CM .

The solution of the eigenvalue problem comes easily when one uses the following identity for the Pauli matrices -as can be directly tested out by applying -→ σ i • -→ σ j on the (symmetric) spin one state and on the (antisymmetric) spin zero state:

-→ σ i • -→ σ j = -1 + 2P i<->j (8)
where P i<->j is the operator that permutes the spin states of the two particles i and j. One sees at once that the Hamiltonian is almost diagonal in our Hilbert space.

For states with total spin S = 3/2 the eigenvalue of H CM is then:

8 3 (C 12 + C 13 + C 23 ) (9)
while for the spin S = 1/2 baryons, it reduces to :

8 3 (C 12 -4C 13 ) (10)
for all states with two identical quarks (they are chosen above as being q 1 and q 2 ). The mass of corresponding S=1/2 states reads then:

2m 1 + m 3 + 8 3 (C 12 -4C 13 ) (11)
The only two states that are mixed are the spin 1/2 states where all three flavours are different. Using the same spin coupling scheme as before, where the spin of the first two quarks is coupled to one (ψ 1 ) or zero (ψ 2 ), and where the total spin is 1/2:

ψ 1 = |(q 1 q 2 ) 1 ⊗ |(q 3 ) 1/2 ψ 2 = |(q 1 q 2 ) 0 ⊗ |(q 3 ) 1/2 (12)
one easily finds the colour-spin Hamiltonian over these two states as:

H CM = 8 3   C 12 -2(C 13 + C 23 ) - √ 3 (C 13 -C 23 ) - √ 3 (C 13 -C 23 ) -3C 12   (13)
Now we see that if C 13 = C 23 this matrix is also diagonal. If q 1 is the u-quark and q 2 is the d-quark, we would expect this to be approximately true. The same remark holds for the effective masses m 1 ≃ m 2 . Then we note that in this approximation the largest eigenvalue leads to a mass:

2m 1 + m 3 + 8 3 (C 12 -4C 13 ) ( 14
)
and that is the mass of both the uuq 3 and ddq 3 systems given by expression (11). Therefore we have three states in the octet made from quarks u, d and q 3 (q 3 being s, c or b), with the same mass ( as they would have if we had an isospin 1 state and isospin invariance in nature ) .

The approximate isospin independence that sits in the QCD Lagrangiandue to the smallness of the (current) u and d mass compared to Λ QCD ,reappears in the masses of the three-quark bound states.

If we make the baryons out of the lightest quarks u, d and s, we see that the states we called ψ 1 and ψ 2 are those commonly denoted Σ 0 and Λ.

The mixing between ψ 1 and ψ 2 -in this case called Λ -Σ 0 mixing -is induced by an ( isospin breaking ) inequality C 13 = C 23 .

So this type of mixing is the general one for all ψ 1 and ψ 2 states, it is quite small for uds, udc, udb, much bigger for states like ucb.

In the general case the eigenvectors for H cm are :

V + = cosθ • ψ 1 + sinθ • ψ 2 V -= -sinθ • ψ 1 + cosθ • ψ 2
with corresponding eigenvalues for H cm :

λ ± = 8 3 [-(C 12 +C 13 +C 23 ) ± 2 C 12 2 -C 12 C 13 -C 12 C 23 + C 13 2 -C 13 C 23 + C 23 2 ]
and mixing angle θ given by tan (2

• θ) = - √ 3 • (C 13 -C 23 ) 2 • C12 -(C13 + C23 ) ( 15
)
For light (q = u, d and s) quarks, baryon masses are well reproduced with masses

m q = 360, m s = 535 MeV, (16)
C ss = 10, C cs = 4, C cc = 4 MeV. ( 19
)
The states we have constructed are considerably simpler than those one often6 finds in textbooks and lecture notes. No mention is here of "mixed symmetry octet states" or "generalized Pauli principle". The states are symmetrized only where the Pauli principle demands it, and can be used for all calculations where flavour is not changed.

can be obtained by using as input the observed nucleon and Λ values, i.e.

µ u = µ d = µ P , µ s = -3 • µ Λ .
The student will remark that µ s < µ u as expected, but also that not all is well with this kind of calculation. An obvious problem is that experimentally µ Λ -µ Ξ -is positive, whereas this conventional type of calculation gives µ Λ -µ Ξ -= (µ s -µ d )/9 which should be negative!

We now turn to interactions that change flavour, and we shall see that permutation symmetry reestablishes something equivalent to the "generalized Pauli principle" without invoking any "principle", just the consistency of the Hilbert space.

5 Flavour changing forces and the emergence of the generalized Pauli principle.

Suppose now that we want to teach flavour changing weak decays, as it is the case in semi-leptonic hyperon decays. Let us take the example of an s-quark turning into a u-quark as in the Λ → P e -ν decay.

The state Λ is constituted by three quarks of different flavours: u, d, and s, so the Pauli principle does not bring any constraint in this case. This resonance has a total spin S = 1/2, and its (ud) part is of spin S = 0 when we ignore ΛΣ 0 mixing. We are therefore inclined to write it as:

Λ = (u ↑ d ↓ -u ↓ d ↑ )s ↑ / √ 2
In the transformation s → u, the above expression changes into:

(u ↑ d ↓ -u ↓ d ↑ )u ↑ / √ 2
and this state is not in the Hilbert space defined in the previous section! It is not a state allowed by the Pauli principle.

The result is a state vector where the first term is symmetric under the interchange of the two u-quarks -and that is fine-but the second term is not! As a consequence, our Hilbert space is not appropriate to admit the action of Hamiltonian corresponding to the weak |Λ decays.

Looking at the list of semi-leptonic decays of low-lying spin 1/2 baryons, one realizes that the same kind of pathology is present as soon as the state |abc is transformed into the state |aac , or inversely when the state |aac is transformed into a combination of |abc states, with a, b and c figuring out the quark flavours. But we see that this problem can easily be cured in the above considered |Λ decay by choosing for |Λ a wave function symmetric in u and s :

Λ = (u ↑ d ↓ s ↑ + s ↑ d ↓ u ↑ -u ↓ d ↑ s ↑ -s ↑ d ↑ u ↓ )/2
and in the general case by imposing the baryon wavefunctions to be completely symmetric in all the three quarks whether they were identical or not. The necessity of having a "complete" Hilbert space is what leads to the "generalized" Pauli principle.

Actually, it can be shown that the Hilbert space description in terms of completely symmetrized states in the three quarks is equivalent to the description in terms of states where the symmetry is imposed only for quarks of the same flavour. In other words, there is an isomorphism between the physics described in terms of two Hilbert spaces, the first one submitted to

Conclusion

We have shown that the classification of baryons, and reasonable estimates of their masses, their magnetic moments, as well as of the corresponding form factors relative to semi-leptonic decays can be obtained without any explicit reference to the flavour unitary group. All this can be included as well in fairly elementary quantum mechanics courses, as in elementary particle courses.

As widely explicated, the "minimal" or "partial" permutational symmetry imposed on wave-functions by the Pauli principle is well adapted as long as flavour changing forces do not operate. When they do, one is naturally led to the choice of wave functions satisfying complete symmetry under permutation of the three constituent quarks, that is, in other words, to the application of the generalized Pauli principle.

At this point it is interesting to make a connection between the above discus- as follows: (S=1/2; 8) (S=3/2; 10). And we can here recognize the decomposition of the completely symmetric 56 representation of the so-called flavour-spin SU(6) group commonly used to classify the S-wave baryons (in perfect accordance with results of Section 2). That also makes unitary groups so natural candidates for (broken) symmetry groups.

where: M = m 1 + m 2 + m 3 , and two non zero eigenvalues λ ± respectively associated to the eigenvectors:

-→ ρ = h + (m 1/2 1 -→ r 1 + m 1/2 2 f + -→ r 2 + m 1/2 3 g + -→ r 3 ) (22)
and:

-→ λ = h -(m 1/2 1 -→ r 1 m 1/2 2 f --→ r 2 + m 1/2 3 g --→ r 3 ) (23)
with:

h ± = λ ± (3K) 1 2 (1 + f 2 ± + g 2 ± ) K = k 12 k 13 + k 21 k 23 + k 31 k 32
and:

f ± = m 2 m 1 [k 23 + k 31 -m 3 λ ± )][k 31 + k 12 -m 1 λ ± ] -k 2 31 k 23 k 31 + k 12 [k 23 + k 31 -m 3 λ ± ] g ± = √ m 3 k 23 [- k 12 √ m 1 + k 23 + k 12 -m 2 λ ± √ m 2 f ± ]
Finally, the explicit expressions of λ ± read:

λ ± = 1/2 [ i; i =j =k =i k ij + k ik m i ±   i; i =j =k =i k ij + k ik m i   2 -4K j<k 1 m j m k ]
In terms of --→ R cm , -→ ρ , -→ λ , the total Hamiltonian acquires the following simple form in which the c.m. motion explicitly separates:

H h.o. = i m i + p 2 Rcm 2M + p 2 ρ 2m + + p 2 λ 2m - + √ 3K 2 ( -→ ρ 2 + -→ λ 2 ) (24)
where:

m ± = √ 3K λ ±
It might be useful to consider the special case with only one coupling constant, that is: k ij = k for i,j= 1,2,3. Then, K becomes = 3k 2 and, in the Hamiltonian, the potential part reduces to:

3 2 k( -→ ρ 2 + -→ λ 2
). In the kinetic part, one gets: m ± = 3k λ ± and :

λ ± = k [ i 1 m i ± i 1 m 2 i - i<j 1 m i m j ]
while f ± and g ± simplify as :

f ± = m 2 m 1 4[1 -m 3 (λ ± /2k)][1 -m 1 (λ ± /2k)] -1 3 -2m 3 (λ ± /2k) g ± = √ m 3 [ - 1 √ m 1 + 2 1 -m 2 (λ ± /2k) √ m 2 f ± ]
Another relevant case is the one with two identical quarks, that is 1 = m 2 = m and m 3 = m ′ . Then it is reasonable to have : k 31 = k 23 = k ′ and k 12 = k, and the -→ ρ and -→ λ Jacobi vectors simply become:

-→ ρ = 1 √ 2 [ 2k + k ′ 3k ′ ] 1 4 ( -→ r 1 --→ r 2 ) ( 25
) -→ λ = 1 √ 2 [ k ′ 3(2k + k ′ ) ] 1 4 ( -→ r 1 + -→ r 2 -2 -→ r 3 ) (26)
. Finally, we give the limit case with all three masses equal

m 1 = m 2 = m 3 = m and k 12 = k 23 = k 31 = k : -→ ρ = 1 √ 2 ( -→ r 1 --→ r 2 ) (27)
-→ λ = 1 √ 6 ( -→ r 1 + -→ r 2 -2 -→ r 3 ) (28)
We start with some definitions :

Let k be a commutative ring, E a k-module, and N the set of integers.

Def .1: Symmetric tensor algebra:

Denoting T n (E) the set of elements: z = x 1 ⊗ x 2 ⊗ . . . ⊗ x n with x 1 , . . . , x n ∈ M , the action of S n on T n (E) reads:

σ(x 1 x 2 ⊗ . . . ⊗ x n ) = x σ -1 (1) ⊗ x σ -1 (2) ⊗ . . . ⊗ x σ -1 (n) (29)
The elements z such that:

σ • z = z
are called symmetric tensors of order n. They form a sub-k -module of T n (E), denoted: T S n (E). One sets:

T S(E) = ∞ n=0 T S n (E)
on which one can define a symmetric product (we do not give here the rule in order not to overload the text).

Def .2: Gamma algebra:

We call Gamma algebra of E and denote by Γ(E) the associative, unifier, commutative, algebra defined by the set of generators N × E and relations:

(0, x) = 1 (30) (p, λx) = λ p (p, x) (31)
(p, x + y) = p q=0 (p, x)(p -q, y) (32)
Def .3: Exponential type sequence:

We call exponential type sequence of E elements a sequence a = (a p ) p∈N such that

a 0 = 1 (33) a p a q = (p + q)! p!q! a p+q (34)
Now, we come to the results we need:

Prop.1: {(p, x)} p∈N with x ∈ E is an exponential type sequence.

An interesting example of exponential type sequence is given by f

(x) with f (x) p = 1 p! (x) p (35)
Prop.2: There exists one and only one isomorphism g between Γ(E) and T S(E):

Γ(E) ∼ = T S(E) (36)
such that:

(g • γ p )(x) = x ⊗ x ⊗ . . . ⊗ x (37)
with p x-factors, and where we have denoted by γ p , with p ∈ N , the application from E into Γ(E) product of the injection x → (p, x) and of the canonical homomorphism of the free commutative algebra N × E to Γ(E).

Prop.3: Let E and F be two k modules. There exists one and only one

isomorphism φ from Γ(E × F ) into Γ(E) ⊗ k Γ(F ): Γ(E × F ) ∼ = Γ(E) ⊗ k Γ(F ) (38)
such that:

(φ • γ p )(x, y) = p q=0 γ q (x) ⊗ γ p-q (y) (39)
The above properties can be reformulated by considering the case of polynomials. Indeed, one directly remarks that Γ(E) can be seen as the set of polynomials on the dual E * of E, that is also:

Γ(E) ∼ = P olyn(E * ) (40)
Now, considering E finite dimensional, that is made of elements -→ x = i x i e i ∈ E where (e i ) i∈I is a basis of E, with x i ∈ k for each i ∈ I, and I being finite, then Prop.2 insures that:

Γ(E) ∼ = P olyn(E * ) ∼ = (P olyn(E)) * ∼ = T S(E) (41)
and from Prop.3 we get:

P olyn( -→ x , -→ y ) = P olyn( -→ x ) ⊗ P olyn( -→ y ) (42)
with -→ x ∈ E and -→ y ∈ F , keeping in mind the relations (31),( 34) and (38).

Coming back to our flavoured quarks, let us associate to each flavour f = u, d,. . . the S = 1/2 spin representation space E f generated by f ↑ , f ↓ . One can write:

Γ(E f ) ∼ = P olyn(f ↑ , f ↓ ) (43)
and also :

Γ(E f × E f ′ ) ∼ = Γ(E f ) ⊗ R Γ(E f ′ ) (44)
our states being defined on the ring of real numbers R.

Appendix C: Fock space wave-functions Actually, the above developed framework is nothing else than Fock space.

Let us associate to each just above defined couple f ↑ , f ↓ in E f the -commuting or "bosonic" -creator operators a † f ↑ , a † f ↓ , acting on the |0 vacuum. Again, we can write:

Γ(E f ) ∼ = P olyn(a † f ↑ , a † f ↓ ) (45)
and also:

P olyn(a † f ↑ , a † f ↓ ) ⊗ P olyn(a † f ′↑ , a † f ′↓ ) ∼ = P olyn(a † f ↑ , a † f ↓ , a † f ′↑ , a † f ′↓ ) (46)
Due to the invariance under permutations of a monomial in a † f operators, one can get, for baryon states, expressions as simple as the ones used in Section (4) and denoted in Appendix D below "simply Pauli symmetric" -so much simpler than those called "generalized Pauli symmetric" in the same appendix -but unambiguously defined, as these last ones, for computations involving flavour changes. More precisely, it is rather immediate to translate a wave-function written in our "Pauli symmetric basis" into the corresponding one in the Fock basis. One has just to take care of the respective normalization of states: as an example, the spin S = 1, S z = 0 state (u ↑ u ↓ + u ↓ u ↑ )/ √ 2 will correspond to a † u ↑ a † u ↓ , while u ↑ u ↑ will be associated to a † u ↑ a † u ↑ / √ 2, using the usual normalized power product (a † f ) n / √ n!. An explicit comparison of the wave-functions for S = 1/2 low-lying baryons in the different bases can be performed with Appendix D.

Finally, associating, as usual, to each creator a † f its annihilator counter part a f such that a f |0 = 0 and (a † f ) † = a, and satisfying the well-known Heisenberg algebra commutation relations:

[a i , a † j ] = δ ij (47)
[a i , a j ] = [a † i , a † j ] = 0 (48) with i,j representing any f ↑ orf ↓ , simple expressions can be given for the self-adjoint parts of the Hamiltonian representing the flavour changing f into f ′ :

H ± f →f ′ = (a † f ′↑ a f ↑ ± a † f ′↓ a f ↓ + h.c.) (49)
involved in the computations of the vector f 1 and axial g 1 form factors of baryon semi-leptonic decays.

It might look surprising to the reader that we use a bosonic Fock space although we are dealing with fermionic quarks. We should not forget that antisymmetry is carried by the colour part. In other words, let us say that in the fermionic space of coloured quarks, there exists a subspace constituted by three quark states which are colour singlets -our baryons -this subspace being isomorphic, as a Hilbert space, to the three particles subspace of a bosonic Fock space.

, Appendix D: Wave-functions of spin 1/2 S-wave baryons

Different expressions for the wave-functions of baryons have been used following the kind of computations we had to consider. As long as classification and flavour non changing quantities are involved, wave-functions satisfying the symmetry imposed by the Pauli principle are perfectly adapted: we will call them "partial symmetric" or "simple Pauli symmetric" wave-functions.

As soon as flavour changing forces are involved, complete permutational invariance of the three quarks is required : we will denoted them "generalized Pauli symmetric" wave-functions. In this last case, the use of the Fock space formalism, as seen in Appendix C, looks rather appealing: their natural denomination will be "Fock" wave-functions. Let us once more mention that the colour part, and so the antisymmetry nature of the baryon, has been "factored out", that is why one is concerned below only with permutational symmetry aspects, and in particular with bosonic Fock operators.

Simple Pauli symmetric wave-functions

N = [2d ↑ d ↑ u ↓ -(d ↑ d ↓ + d ↓ d ↑ )u ↑ ]/ √ 6 P = [2u ↑ u ↑ d ↓ -(u ↑ u ↓ + u ↓ u ↑ )d ↑ ]/ √ 6 Λ = [(u ↑ d ↓ -u ↓ d ↑ )s ↑ ]/ √ 2 Σ O = [2u ↑ d ↑ s ↓ -(u ↑ d ↓ + u ↓ d ↑ )s ↑ ]/ √ 6 Σ + = [2u ↑ d ↑ s ↓ -(u ↑ u ↓ + u ↓ u ↑ )s ↑ ]/ √ 6 Σ + = [2d ↑ d ↑ s ↓ -(d ↑ d ↓ + d ↓ d ↑ )s ↑ ]/ √ 6 Ξ -= [2s ↑ s ↑ d ↓ -(s ↑ s ↓ + s ↓ s ↑ )d ↑ ]/ √ 6 Ξ 0 = [2s ↑ s ↑ u ↓ -(s ↑ s ↓ + s ↓ s ↑ )u ↑ ]/ √ 6
Generalized Pauli symmetric wave-functions

N = [2d ↑ d ↑ u ↓ -(d ↑ d ↓ + d ↓ d ↑ )u ↑ + permutations]/ √ 18 P = [2u ↑ u ↑ d ↓ -(u ↑ u ↓ + u ↓ u ↑ )d ↑ + permutations]/ √ 18 Λ = [(u ↑ d ↓ -u ↓ d ↑ )s ↑ + permutations]/ √ 12

  oscillator with two quark spin one or by the -→ λ oscillator with two quark spin zero, and total spin S = 1/2. ii) Baryons made of two (and only two) identical flavour quarks qqq ′ : Now the Pauli principle imposes permutation symmetry only in the first two quarks qq. Note that, as explicated in Appendix A, the antisymmetry ( resp. symmetry ) of -→ ρ ( resp. -→ λ ) persists when m 1 = m 2 = m and m 3 = m ′ with m and m ′ different. Then one gets:

(

  and partial qq spin S = 0) or from the -→ λ osc. ( and partial qq spin S = 1). There are also 6 states of qqq'-type made from the -→ ρ osc. ( and partial qq spin S = 0), 6 states of qqq ′ -type made from the -→ λ osc. ( and partial qq spin S = 1). Finally, there are 4 states of qq ′ q"-type respectively made with the -→ ρ and -→ λ osc. and of qq partial spin S = 0 and 1. Then, with three flavours, one obtains a total of 70 states, with 8 states of S = 3/2 and 19 states of S = 1/2 in perfect accordance with the usual SU(6) approach where the P-wave baryons ( L= 1 ) are classified in the irreducible 70 dimensional SU(6) representation, itself decomposing with respect to SU(3) flavour and SU(2) spin as: 70 = (8, 3/2) + (8 + 10 + 1, 1/2)

  and strength factors C qq = 18.5, C qs = 12.5, C ss = 9.5 MeV. (17) If one accepts that effective masses could differ when heavy quarks are present, a possible choice for S-wave charmed baryon states could be m c = 1550, m q = 450, m s = 590 MeV, (18) with associated strength factors C qq = 20, C qc = 5, C qs = 15,

  the usual Pauli principle and the second one submitted to what has been called the "generalized " Pauli principle. The mathematical proof is sketched in Appendix B. Let us point out that, instead of the commonly adopted representation for the baryon wave-function ( we call it "Generalized Pauli symmetric" in Appendix D ), one might prefer the equivalent, simpler and more elegant expressions offered by the Fock space formalism, as developed in Appendix C and explicated in Appendix D for the octet of S = 1/2 low-lying baryons. Finally, combining the completely antisymmetric colour part of the three quark wave functions with its completely symmetric complementary part provides, of course, with a wave function which is completely antisymmetric in the three constituent fermions, and that is namely what is prescribed by the generalized Pauli principle.

  sion in Section 5 and the classification of baryons in irreducible representions of the SU(N) group, N being the number of different flavours. Actually, one knows that each irreducible representation R of the SU(N) group can be represented by a Young tableau with n boxes associated with an irreducible representation of the symmetric group S n also called permutation group P n of n elements. That means that any element of R satisfies a special P n symmetry which is the same for all elements in R. Taking as an example SU(2), the representations above denoted S=3/2 and S=1/2 resulting from the tensorial product of three times the fundamental representation S=1/2 are respectively completely symmetric and "mixed symmetric" under P 3 . Now, considering the group SU(3), its irreducible eight dimensional 8 and ten dimensional 10 representations obtained by tensorial product of three fundamental representations 3 are respectively completely and mixed symmetric under P 3 . Complete symmetry under flavour and spin will therefore be obtained by combining the spin-SU(2) and flavour-SU(3) representations

As A

3/2 dominates at low energy, the analysis of data were quite simple.

We do not consider mixing with multiquark states. For mesons we need a generalization of the Pauli principle which follows from local field theory through the CPT operation; but this point will not be discussed hereafter where we limit to (usual) baryons.

The following arguments are not restricted to nonrelativistic quantum mechanics.

If we added on HS =i,j cij -→ σ i • -→ σ j we would get the most general Hamiltonian we can have for the system of three quarks when the spatial variable are integrated out. So H is more general than one gluon exchange only.

But not always !

In this case our notation is not optimal. The coefficients Cij should then be labeled by the quark flavours and not by numbers.

Acknowledgements :

P.S. is indebted to P. Aurenche and E. Guadagnini for numerous and valuable discussions. We both must specially thank R. Stora for his precious help in some algebraic aspects developed in this note, particularly in Appendices B and C, as well as for a critical reading of the manuscript.

If one wants to complicate calculations by symmetrizing in all three flavours, one is of course free to do so 7 .

Mathematically there is indeed a one to one correspondence between the states that we have used and the states that are symmetrized in all three particles. But to ask a student to compute the magnetic moment of the nucleons by using nine terms in the state, instead of three, is not very kind.

That calculation is very simple indeed, using the magnetic moment operator for a baryon:

where in the definition of µ i we have taken out the electric charge factor Q i of the quark i, but not the expected 1 m i •c dependence.All the µ i 's are therefore positive.

The expectation value of -→ µ for a |B, S = 1/2, S z = 1/2 state, composed of two identical quarks α and a different one β gives the value of the magnetic moment of the baryon B:

With these conventions one gets for the proton and neutron:

9 µ d and so on. In the case where all three flavours are different, q 1 = α, q 2 = β and q 3 = γ, one finds for the two states

We include the off-diagonal matrix element for those that want to compute the effect of ψ 1 ψ 2 mixing:

Neglecting small mixing, the magnetic moment of Λ is given as µ Λ = -1 3 µ s . A tolerable estimation of all the magnetic moments of the lightest baryons

Appendix A: Jacobi coordinates

The spatial wave functions ψ of our problem are the harmonic-oscillator eigenfunctions ψ nLM ( -→ ρ , -→ ), where -→ ρ and -→ λ are the Jacobi coordinates which separate the Hamiltonian in Eq. (20) into two independent threedimensional oscillators.

Hereafter, we give expressions for Jacobi coordinates that makes the separation of the center of mass motion in the case of unequal masses, and even when the interaction potential is different for each pair of particles.

So, starting from the Hamiltonian:

with each coupling k ij = k ji explicitly depending of the quark pair q i q j , one can rewrite the potential part as:

and -→ y i defined as: -→ y i = √ m i -→ r i , and the entries of the matrix M satisfying:

The diagonalization of M provides with one eigenvalue λ 0 = 0 associated to an eigenvector proportional to:

Appendix B: Generalized Pauli principle versus Pauli principle: a mathematical proof of their equivalence This section is not directly dedicated to students. Of course, one can try and convince oneself on some examples that there is a one-to-one correspondence between the "mathematical" states and observables submitted to the Generalized Pauli principle (or GPP) and those simply satisfying the Pauli principle (or PP). The proof that we propose hereafter has two advantages.

First, it is general, and so adaptable to situations other than that of regular baryons (for example multiquark states). Secondly, it introduces the Fock space formalism in a rather natural way. All that follows is taken from [3].


Anh Tuan Bui

Alexander D Rahm

Matthias Wendt

THE FARRELL-TATE AND BREDON HOMOLOGY FOR PSL 4 (Z) VIA CELL SUBDIVISIONS

Keywords:

We provide some new computations of Farrell-Tate and Bredon (co)homology for arithmetic groups. For calculations of Farrell-Tate or Bredon homology, one needs cell complexes where cell stabilizers fix their cells pointwise. We provide two algorithms computing an efficient subdivision of a complex to achieve this rigidity property. Applying these algorithms to available cell complexes for PSL 4 (Z) provides computations of Farrell-Tate cohomology for small primes as well as the Bredon homology for the classifying spaces of proper actions with coefficients in the complex representation ring.

Introduction

Understanding the structure of the cohomology of arithmetic groups is a very important problem with relations to number theory and various K-theoretic areas. Explicit cohomology computations usually proceed via the study of the actions of the arithmetic groups on their associated symmetric spaces, and recent years have seen several advances in algorithmic computation of equivariant cell structures for these actions. To approach computations of Farrell-Tate and Bredon (co)homology of arithmetic groups, one needs cell complexes having a rigidity property: Cell stabilizers must fix their cells pointwise. The known algorithms (using Voronoi decompositions and such techniques, cf. e.g. [START_REF] Mathieu Dutour Sikirić | On the cohomology of linear groups over imaginary quadratic fields[END_REF][START_REF] Elbaz | Perfect forms, K-theory and the cohomology of modular groups[END_REF]) do not provide complexes with this rigidity property, and both for the computation of Farrell-Tate cohomology (resp. the torsion at small prime numbers in group cohomology) of arithmetic groups as well as for the computation of Bredon homology, this lack of rigid cell complexes constitutes a significant bottleneck.

In theory, it is always possible to obtain this rigidity property via the barycentric subdivision. However, the barycentric subdivision of an n-dimensional cell complex can multiply the number of cells by (n + 1)! and thus easily let the memory stack overflow. We provide two algorithms, called Rigid Facets Subdivision, cf. Section 2, and Virtually Simplicial Subdivision, cf. Section 3, as well as a combination of them (Hybrid Subdivision, cf. Section 4) which subdivide cell complexes for arithmetic groups such that stabilizers fix their cells pointwise, but only lead to a controlled increase (in terms of sizes of stabilizer groups) in the number of cells, avoiding an explosion of the data volume. An implementation of the algorithms, cf. [START_REF] Bui | Torsion Subcomplexes Subpackage, version 2.1, accepted sub-package in HAP (Homological Algebra Programming) in the computer algebra system GAP[END_REF], shows that cases like PSL 4 (Z) can effectively be treated with it, using commonly available machine resources. For the sake of comparison, barycentric subdivison applied to the cell complex for PSL 4 (Z) from [START_REF] Mathieu | On the integral homology of PSL 4 (Z) and other arithmetic groups[END_REF] would produce 3540 times as many top-dimensional cells as Rigid Facets Subdivision does -see Table 1.

1.1. Computations of Farrell-Tate cohomology. Farrell-Tate cohomology is a modification of cohomology of arithmetic groups which is particularly suitable to investigate torsion related to finite subgroups (in particular, the torsion in cohomological degrees above the virtual cohomological dimension). While the known cell complexes for arithmetic groups can deal very well with the rational cohomology and torsion at primes which do not divide orders of finite subgroups, computations with these complexes run into serious trouble for small prime numbers because the differentials in the relevant spectral sequence are too complicated to evaluate. There is a suitable new technique called torsion subcomplex reduction, cf. [START_REF] Rahm | Accessing the cohomology of discrete groups above their virtual cohomological dimension[END_REF], which produces significantly smaller cell complexes and therefore simplifies the equivariant spectral sequence calculations. To apply this simplification, however, one needs cell complexes with the abovementioned rigidity property. Applying Rigid Facets Subdivision to a cell complex for PSL 4 (Z), we have computed the Farrell-Tate cohomology of PSL 4 (Z), at the primes 3 and 5. These results can be found in Theorem 10 and Proposition 14. Since the Table 1. Numbers of cells in the individual dimensions of the studied non-rigid fundamental domain X for PSL 4 (Z) from [START_REF] Mathieu | On the integral homology of PSL 4 (Z) and other arithmetic groups[END_REF], its rigid facets subdivsion RFS(X), its virtually simplicial subdivision VSS(X), its hybrid subdivision HyS(X) and its barycentric subdivision BCS(X). In the cases which are effectively of rank one (5-torsion in PSL 4 (Z)), we can check the results of the cohomology computation using torsion subcomplex reduction by comparing to a computation using Brown's formula.

Computations of Bredon homology.

For any group G, Baum and Connes introduced a map from the equivariant K-homology of G to the K-theory of the reduced C * -algebra of G, called the assembly map. For many classes of groups, it has been proven that the assembly map is an isomorphism; and the Baum-Connes conjecture claims that it is an isomorphism for all finitely presented groups G (counter-examples have been found only for stronger versions of the Baum-Connes conjecture). The assembly map is known to be injective for arithmetic groups. For an overview on the conjecture, see the monograph [START_REF] Mislin | Proper group actions and the Baum-Connes conjecture[END_REF].

The geometric-topological side of Baum and Connes' assembly map, namely the equivariant Khomology, can be determined using an Atiyah-Hirzebruch spectral sequence with E 2 -page given by the Bredon homology H Fin n (EG; R C ) of the classifying space EG for proper actions with coefficients in the complex representation ring R C and with respect to the system Fin of finite subgroups of G. This Bredon homology can be computed explicitly, as described by Sanchez-Garcia [START_REF] Rubén | Bredon homology and equivariant K-homology of SL(3, Z)[END_REF][START_REF] Rubén | Equivariant K-homology for some Coxeter groups[END_REF].

While for Coxeter groups with a small system of generators [START_REF] Rubén | Equivariant K-homology for some Coxeter groups[END_REF] and arithmetic groups of rank 2 [START_REF]On the equivariant K-homology of PSL 2 of the imaginary quadratic integers[END_REF], general formulae for the equivariant K-homology have been established, the only known higherrank case to date is the example SL 3 (Z) in [START_REF] Rubén | Bredon homology and equivariant K-homology of SL(3, Z)[END_REF]. Although there are by now considerably more arithmetic groups for which cell complexes have been worked out [START_REF] Mathieu | On the integral homology of PSL 4 (Z) and other arithmetic groups[END_REF][START_REF] Mathieu Dutour Sikirić | On the cohomology of linear groups over imaginary quadratic fields[END_REF][START_REF] Elbaz | Perfect forms, K-theory and the cohomology of modular groups[END_REF], no further computations of Bredon homology H Fin n (EG; R C ) have been done since 2008 because the relevant cell complexes fail to have the rigidity property required for Sanchez-Garcia's method. We discuss an explicit example, cf. Section 10, demonstrating that the rigidity property is essential for the computation of Bredon homology and cannot be circumvented by a different method.

Applying rigid facets subdivision to the cell complex for EPSL 4 (Z) from [START_REF] Mathieu | On the integral homology of PSL 4 (Z) and other arithmetic groups[END_REF], we obtain

H Fin n (EPSL 4 (Z); R C ) ∼ =                0, n 4, Z, n = 3, 0, n = 2, Z 4 , n = 1, Z 25 ⊕ Z/2, n = 0.
Organization of the paper. In Section 2, we provide the Rigid Facets Subdivision algorithm. In Section 3, we provide the Virtually Simplicial Subdivision algorithm. In Section 8, we apply the Rigid Facets Subdivision algorithm to a PSL 4 (Z), and compare the result with a computation using Brown's conjugacy classes cell complex. Finally in Section 10, we provide a counterexample in order to contradict the possibility to compute the Bredon homology from an arbitrary non-rigid cell complex.

Algorithm 1 Subdivide to get stabilizers which fix their cells pointwise

Input: An n-dimensional Γ-equivariant CW-complex X with finite cell stabilizers and a metric as in Remark 2. Output: A rigidification of X (that is, an equivalent Γ-cell complex on which each stabilizer fixes its cell pointwise).

for m running from 1 to n do for σ running through lifts of m-cells in Γ \X (m) do if σ is not rigid then Use Algorithm 2 or 4 to subdivide σ into a partition P , which is a union of rigid m-cells, disjoint up to boundaries, with a fundamental domain F for the Γ σ -action on P . Run through all the (m + 1)-cells; if their boundaries contain σ, then replace σ by its partition P . Replace the cell σ by F in Γ \X (m) . end if end for end for Definition 1. Following the notation in [START_REF] Bui | Computing Bredon homology of groups[END_REF], we use the term Γ-equivariant CW-complex, or simply Γ-cell complex, to mean a CW-complex X on which a discrete group Γ acts cellularly, i.e., in such a way that the action induces a permutation of the cells of X. We say the cell complex is rigid if each element in the stabilizer of any cell fixes the cell pointwise.

Remark 2. The algorithm producing the subdivision of the Γ-equivariant CW-complex X only modifies combinatorial data, based on the barycentric subdivison of individual cells. We require X to come with a geometric realization, equipped with a metric such that each of the cells of X is convex, the restriction of the metric to each cell is CAT(0) and the cell stabilizers act by CAT(0)-isometries. We are not requiring that the metric is CAT(0) on the whole CW-complex. Note however that the examples we are most interested in, are those where Γ is an arithmetic group and the geometric realization of X is the associated symmetric space. Definition 3. A rigidification X of a Γ-cell complex X is a rigid Γ-cell complex X with the same underlying topological space as X. The map passing through the underlying topological space is then a Γ-equivariant homeomorphism X → X of Γ-spaces. Note that a Γ-equivariant homeomorphism of Γ-spaces does not need to preserve existing cell structure, so X is allowed to have more cells than X.

The outer shell of the rigid facets subdivision is Algorithm 1, which subdivides (whenever necessary) representatives of cell orbits using Algorithm 2, respectively Algorithm 4. Proposition 4. Let Γ be a discrete group, and let X be a Γ-equivariant CW-complex having finitely many Γ-orbits and finite cell stabilizers. Assume furthermore that X is equipped with a metric as in Remark 2. Then Algorithm 1 finds a rigidification of X (with respect to the Γ-action). It terminates in finite time.

Proof. The key step of the algorithm is proved by Lemma 7 below; the rest is a routine induction. Lemma 7 is the point where the convexity and isometry requirements are needed. By the finiteness assumptions for orbits and cell stabilizers, the loops are all deterministic over finite index sets. Each operation inside them takes finite time, cf. Corollary 8, whence the claim.

Observation 5. The outer shell Algorithm 1 can be used with any subdivision algorithm for the cells. In particular, replacing the use of Algorithm 2 by the barycentric subdivision in Algorithm 1, the claims of Proposition 4 still hold. However, as mentioned in the introduction, the reason for developing Algorithm 2 is to reduce the blow-up in the number of cells, so as to make the algorithm practically applicable to cell complexes for higher-rank arithmetic groups.

We now discuss the actual subdivision to rigidify cells, Algorithm 2.

Algorithm 2 -Rigid Facets Subdivision

Input: An m-cell σ with stabilizer group Γ σ , with rigid faces, equipped with a metric as in Remark 2. Output: A Γ σ -equivariant set of rigid m-cells, disjoint up to boundaries, constituting a partition P of σ, together with a contractible fundamental domain F (a single m-cell) for the action of Γ σ on P .

• Sort the (m -1)-faces of σ into orbits {{g jt σ j } t } j under the action of Γ σ , where j is indexing the orbits and t is indexing the cells inside each orbit. • Let T be the list containing the element g 11 σ 1 . while #T < #{orbits of (m -1)-faces of σ} do Choose one cell τ ∈ {g jt σ j } in the next orbit {g jt σ j } not yet represented in T , if the union of τ with the cells in T has vanishing naive Euler characteristic then if {τ } ∪ T is contractible then Construct the boundary S of {τ } ∪ T . if S has the naive Euler characteristic of an (m -2)-sphere then if #T < #{orbits of (m -1)-faces of σ}-1 then Add the chosen cell τ to T . else if S is simply connected then Add the chosen cell τ to T .

end if end if end if end if end if end while • Use Algorithm 3 to construct the m-cell F := | τ ∈T convex envelope(τ , barycenter(σ))|.
• Let Γ pw σ be the subgroup of Γ σ which fixes the cell σ pointwise.

• Then P := 1 t |Γσ/Γ pw σ | g 1t F
is the desired partition of σ. • Return P and F . Remark 6. A) Essentially, Algorithm 2 produces a convex union of cells of the barycentric subdivision which is a fundamental domain for the Γ σ -action. The slight complications arise from the fact that Figure 1. From left to right: 1) A full square with natural cell structure and action by its stabilizer mirroring it onto itself. 2) Barycentric subdivision of the square. 3) Rigid facet subdivision of the square with respect to the mirroring action. 4) Virtually Simplicial Subdivision of the square with respect to the mirroring action; note that the fundamental domain F can be chosen arbitrarily by selecting one cell among each of the pairs {a 1 , a 2 }, {b 1 , b 2 } and {c 1 , c 2 }.

c 2 c 1 b 2 a 2 b 1 a 1
we do not actually want to compute the full barycentric subdivision, to gain computational feasibility. B) The contractibility check on {τ } ∪ T in Algorithm 2 can be implemented with Ellis' method [START_REF] Ellis | Computational homotopy of finite regular CW-spaces[END_REF].

Lemma 7 (Rigid Facets Lemma). Let σ be a cell (with stabilizer Γ σ ), whose faces are all rigid and which is equipped with a metric as in Remark 2. Let Γ pw σ be the subgroup of Γ σ which fixes the cell σ pointwise. Then there is a fundamental domain F for the action of Γ σ /Γ pw σ on σ such that σ is tessellated by |Γ σ /Γ pw σ | copies of F . Proof. First we have to check that the statement is well defined in the sense that Γ σ /Γ pw σ is a group. This is the case because for all g in Γ σ , for all γ in Γ pw σ , for all x in σ we have (g -1 γg)x = g -1 (γ(gx)) = g -1 (gx) = x. Therefore, as the kernel of the action of Γ σ on σ, Γ pw σ is a normal subgroup; and there is a short exact sequence of groups,

1 → Γ pw σ → Γ σ → Γ σ /Γ pw σ → 1, which makes our statement well defined.
Suppose that α is one of the facets of σ. We are going to prove that the size of the orbit of α, under the action of Γ σ on the set of facets of σ, is |Γ σ /Γ pw σ |. Let Γ α be the stabilizer of α. We claim that Γ α ∩ Γ σ = Γ pw σ . The action of Γ σ on the compact set closure(σ) is by homeomorphisms; therefore, any element of Γ σ fixing σ pointwise also fixes the boundary ∂σ pointwise. Hence Γ α ∩ Γ σ ⊃ Γ pw σ . On the other hand, let g ∈ Γ α ∩ Γ σ . Then by assumption on the rigidity of the facets, g fixes the cell α pointwise. Since the cell σ is convex and the group acts by CAT(0)-isometries, the barycenter of σ preserves its distances to the boundary ∂σ under the action of Γ σ on σ, and hence remains fixed. As a further consequence of the CAT(0) isometry, the fixed point set of g extends, by preservation of the distances, from the convex envelope of α and the barycenter of σ to the whole cell σ. Hence, g is an element of Γ pw σ . Thus, we can conclude that Γ α ∩ Γ σ = Γ pw σ . Whence, the size of the orbit of α under the action of

Γ σ is |Γ σ /Γ pw σ |. Furthermore, from Γ α ∩ Γ σ = Γ pw
σ , we see that Γ σ /Γ pw σ acts freely on the set of facets of σ. So, we can take one arbitrary representative α k for each orbit of facets, to unite to a fundamental domain for Γ σ /Γ pw σ on the set of facets of σ. Taking the convex envelope e k of α k and the barycenter of σ, we get a fundamental domain F := k e k for Γ σ /Γ pw σ on σ. By the above orbit size calculation, it yields the desired tessellation.

Corollary 8. Algorithm 2 terminates after finitely many steps and produces a rigid subdivision of the cell σ, if the boundary of σ admits a contractible fundamental domain T with simply connected boundary.

Proof. The existence of some fundamental domain F for the action σ is guaranteed by Lemma 7. The contractibility of the fundamental domain T and the simply connectedness of its boundary S ensure that merging the union of cells τ ∈T convex envelope(τ , barycenter(σ)) into one cell F can be realized with the boundary construction ∂F = τ ∈T τ ∪ s∈S e(s), where e(s) is the convex envelope of s and the barycenter of σ.

Algorithm 3 Constructing the union of the convex envelopes

Input: A list T of j-cells, connected by adjacency, and an m-cell σ such that all τ ∈ T are faces of σ. Output: A (j + 1)-cell F which has the same underlying topological space as τ ∈T convex envelope(τ , barycenter(σ)).

Record the barycenter of σ as a new vertex, with stabilizer Γ σ . Enumerate the finite set

S := {ρ is (j -1)-cell | ∃! τ ∈ T : ρ ∈ ∂τ }.
Then S contains all the (j -1)-faces ρ of all τ ∈ T such that ρ is not a common face of any two j-cells τ 1 , τ 2 ∈ T .

• For each s ∈ S, take the convex envelope e(s) of s and the barycenter of σ.

• Record e(s) as an oriented j-cell, with boundary

∂e(s) = {s} ∪ ε∈∂s {convex envelope(ε, barycenter(σ))}.
For the stabilizers, we record

Γ e(s) = Γ s ∩ Γ σ and Γ convex envelope(ε,barycenter(σ)) = Γ ε ∩ Γ σ .
The boundary of the (j + 1)-cell F consists of all the j-faces τ ∈ T and e(s) (for all s ∈ S).

Here, we have to take care of which of the newly constructed cells e(s) are on the same Γ σ -orbit.

In order to decide this, we make use of their common vertex, the barycenter of σ : • Identify the orbits e(s 1 ) and e(s 2 ) if and only if γ ∈ Γ σ : γs 1 = s 2 .

• Attribute arbitrary orientations to Γ σ -representatives of the new cells e(s), and spread them on their Γ σ -orbit using the above identifications. • Return the (j + 1)-cell F with boundary

∂F = τ ∈T τ ∪ s∈S e(s)
and stabilizer Γ σ ∩ τ ∈T Γ τ , where the orbits are subject to the above specified identifications.

Virtually Simplicial Subdivision

As a compromise between Rigid Facets Subdivision (RFS) and Barycentric Subdivision (BCS), we can make a simplification of RFS, which we shall call Virtually Simplicial Subdivision (VSS) and detail in Algorithm 4 below. For a cell complex X, VSS(X) is a refinement of RFS(X), and BCS(X) is a further refinement of VSS(X), as illustrated in Figure 1.

Algorithm 4 -Virtually Simplicial Subdivision

Input: An m-cell σ with stabilizer group Γ σ , with rigid faces, equipped with a metric as in Remark 2. Output: A Γ σ -equivariant set of rigid m-cells, disjoint up to boundaries, consituting a partition P of σ, together with a fundamental domain F (a set of m-cells) for the action of Γ σ on P .

• Sort the (m -1)-faces of σ into orbits {{g jt σ j } t } j under the action of Γ σ , where j is indexing the orbits and t is indexing the cells inside each orbit. • Use Algorithm 3 to construct the m-cells F j := convex envelope(g j1 σ j , barycenter(σ)), where j runs through all orbits. • Let Γ pw σ be the subgroup of Γ σ which fixes the cell σ pointwise. • Then P := j 1 t |Γσ/Γ pw σ | g jt F j is the desired partition of σ. • Return P and the set F := {F j } j .

Corollary 9. Algorithm 4 terminates after finitely many steps and produces a rigid subdivision of the cell σ.

Proof. The existence of the fundamental domain F for the action σ is guaranteed by Lemma 7. We do not need any topological properties for F , because its cells already have the structure constructed with Algorithm 3.

Hybrid Subdivision

The bulk of the processing time of Rigid Facets Subdivision (RFS) and Virtually Simplicial Subdivision (VSS) is being spent on the top-dimensional cells. We can save a considerable amount of processing time as compared to RFS if we do not merge the top-dimensional cells. And we can do that by applying RFS until we are in (top minus 1) dimensions, and then achieve the subdivision using VSS on the top-dimensional cells. We will call this process "Hybrid Subdivision (HyS)". It is faster than both RFS and VSS on large cell complexes, and it produces considerably less cells than VSS. We note however that the VSS process on the top-dimensional cells adds a few low-dimensional cells when connecting to the barycenter, so for instance in Table 5, the low-dimensional numbers of cells are slightly higher for HyS than for RFS, rather than being equal.

Comparison of the various subdivision algorithms

As illustrated in Figure 1, all of the cells constructed by Rigid Facets Subdivision are also constructed by barycentric subdivision; and usually, Rigid Facets Subdivision is coarser than barycentric subdivision. However, the worst-case complexity of Rigid Facets Subdivision is not better than that of barycentric subdivision: If X is a single n-simplex with the natural permutation action of the symmetric group Σ n+1 acting on the vertices, then any rigidification will need to produce at least (n + 1)! = #Σ n+1 top-dimensional cells for X. However, the point is that the average cell complex for interesting arithmetic groups has many of its cells already almost rigid and, only very few with maximally possible stabilizer. Therefore, the complexity for the cases of interest is significantly better than that of the barycentric subdivision, as evidenced by Table 1. As we see from Figure 1, Rigid Facets Subdivision is a minimal rigidification only for the top-dimensional cells; in lower dimensions it creates some superfluous cells at the barycenters of facets, and cells connecting those barycenters with the constructed fundamental domain for the boundary of the subdivided cell.

In Table 1, note that we do not have the numbers of orbits for the barycentric subdivision, because the latter was not constructed; only the numbers of cells have been calculated. In such a construction, the boundaries of the 4-cells would span a 1658304 × 2138688-matrix, and even though that matrix is quite sparse, the authors have not tried to store that amount of information on a machine. In retrospective however, considering that the amount of cells that have been constructed with Virtually Simplicial Subdivision comes close to the amount of cells expected for barycentric subdivision, it turns out that this issue would not prevent applying barycentric subdivision (which in the end was not done, because Table 2 was interpreted by the authors as an exponential growth of the runtime).

In contrast, for SL 3 (Z), the barycentric subdivision is almost reached (see Table 3) : In dimensions 0 and 1, the same cell structure is obtained with both subdivision methods, the only difference being that the four tetrahedra which represent the top-dimensional cells, are merged into one polyhedron when passing to the rigid facets subdivision, along three triangles of trivial stabilizers. Note also that Soulé did in [START_REF] Soulé | The cohomology of SL 3 (Z)[END_REF] carry out the barycentric subdivision by hand. This means for the 2-torsion subcomplex of SL 3 (Z), which we will discuss in Section 6, that we can extract it equivalently by hand from [START_REF] Soulé | The cohomology of SL 3 (Z)[END_REF], or by machine after rigid facets subdivision.

In order to benchmark the run-time, in Table 2, we denote the subgroup of SL 3 (Z) consisting of (1) all matrices whose first column agrees with the first standard basis vector modulo 2 by Γ 1 ;

(2) all matrices which are upper triangular modulo 2 by Γ 2 .

Table 3. Numbers of cells in the individual dimensions, of the studied non-rigid fundamental domain X for SL 3 (Z), its rigid facets subdivsion RFS(X), its virtually simplicial subdivision VSS(X), its hybrid subdivision HyS(X) and its barycentric subdivision BCS(X) (the latter three coincide for SL 3 (Z)). From the non-rigid cell complex describing the action of SL 3 (Z) on its symmetric space, provided by Mathieu Dutour Sikirić in [START_REF] Ellis | Homological algebra programming[END_REF], we obtain, after applying Rigid Facets Subdivision, the following 2-torsion subcomplex, in accordance with Soulé's subdivision [START_REF] Soulé | The cohomology of SL 3 (Z)[END_REF].

Dimension 0 1 2 3 # Orbits in X 1 1 2 1 #
stab(M) ∼ = S4 stab(Q) ∼ = D6 stab(O) ∼ = S4 stab(N) ∼ = D4 stab(P) ∼ = S4 N' M' D2 D3 D3 D2 Z/2 Z/2 Z/2 D4 Z/2 D4 D2 Z/2
Here, the three edges N M , N M ′ and N ′ M ′ have to be identified as indicated by the arrows. All of the seven triangles belong with their interior to the 2-torsion subcomplex, each with stabilizer Z/2, except for the one which is marked to have stabilizer D 2 . Using torsion subcomplex reduction (cf. [START_REF] Rahm | Accessing the cohomology of discrete groups above their virtual cohomological dimension[END_REF]), we reduce this subcomplex to

S 4 O D 2 D 6 Q Z/2 S 4 M D 4 S 4 P D 4 D 4 N ′
and then to

S 4 Z/2 S 4 D 4 S 4
which is the geometric realization of Soulé's diagram of cell stabilizers. This yields the mod 2 Farrell cohomology as specified in [START_REF] Soulé | The cohomology of SL 3 (Z)[END_REF].

Also for PSL 4 (Z), Rigid Facets Subdivision allows us to obtain the 2-torsion subcomplex, but because of the complexity the latter (see Table 4), the current implementation of Torsion Subcomplex Reduction does not get it down to manageable size. In upcoming joint work, the authors are going to Table 4. Numbers of cells in the 2-torsion subcomplex for PSL 4 (Z) before reduction, obtained after rigid facets subdivsion of the studied cell complex, sorted into isomorphism types of their stabilizers. Here, G 1 and G 2 are non-trivial group extensions 5. Numbers of cells in the individual dimensions, of the non-rigid fundamental domain X for Sp 4 (Z) described in [START_REF] Macpherson | Explicit reduction theory for Siegel modular threefolds[END_REF] and implemented by Dutor Sikirić [START_REF] Ellis | Homological algebra programming[END_REF], its rigid facets subdivsion RFS(X), its virtually simplicial subdivision VSS(X), its hybrid subdivision HyS(X) and its barycentric subdivision BCS(X). We shortly discuss the results of applying rigid facets subdivision and torsion subcomplex reduction (for odd primes) to the non-rigid cell complex describing the action of Sp 4 (Z) on the associated symmetric space Sp 4 (R)/ U(2), described in [START_REF] Macpherson | Explicit reduction theory for Siegel modular threefolds[END_REF] and implemented by Mathieu Dutour Sikirić in [START_REF] Ellis | Homological algebra programming[END_REF].

1 → A 4 × A 4 → G 1 → Z/2Z → 1 and 1 → G 0 → G 2 → Z/2Z → 1 for 1 → (Z/2Z) 4 → G 0 → Z/3Z → 1 Stabilizer type A 4 G 1 (Z/2Z) 2 S 4 Z/2Z D 8 (Z/2Z) 3 G 2 Z/4Z
The 3-torsion subcomplex for Sp 4 (Z) is a single vertex whose stabilizer group is the group [72, 30] in the GAP SmallGroup library. This group is of the form C 3 ×((C 6 × C 2 ) ⋊ C 2 ) and its mod 3 cohomology is isomorphic to the mod 3 cohomology of C 3 × S 3 . In particular, we get isomorphisms

H i (Sp 4 (Z), F 3 ) ∼ = H i (C 3 × S 3 , F 3 ), i > vcd(Sp 4 (Z)).
The Hilbert-Poincaré series for the mod 3 cohomology of C 3 × S 3 is given by

HP C3 × S3 (T ; 3) = (1 + T )(1 + T 3 ) (1 -T 2 )(1 -T 4 ) .
In [3, (6.7)], the Hilbert-Poincaré-series for Sp 4 (Z) has been computed. In the notation of loc.cit., the difference to the Hilbert-Poincaré-series for C 3 × S 3 above is given by

P (Γ 6 0 ) -P (B ′ ⋊ Z/2) = 1 + T 3 + T 4 + T 5 1 -T 4 - 1 + T + T 3 + T 4 1 -T 4 = T 5 -T 1 -T 4 = -T.
In particular, the Hilbert-Poincaré series for C 3 × S 3 and Sp 4 (Z) agree above the virtual cohomological dimension of Sp 4 (Z) and thus the computation via rigid facets subdivision and torsion subcomplex reduction agrees with the computation of Brownstein and Lee in [START_REF] Brownstein | Cohomology of the symplectic group Sp 4 (Z). I. The odd torsion case[END_REF].

The 5-torsion subcomplex for Sp 4 (Z) is a single vertex whose stabilizer group is C 10 , whose mod 5 cohomology is isomorphic to the mod 5 cohomology of C 5 . This agrees directly with [START_REF] Brownstein | Cohomology of the symplectic group Sp 4 (Z). I. The odd torsion case[END_REF]Corollary 6.3] in degrees above the virtual cohomological dimension.

Table 5 provides an overview of the numbers of cells resp. orbits in the complexes subdivided by rigid facets subdivision and barycentric subdivision, respectively.

Example: Farrell-Tate cohomology of PSL 4 (Z) at the prime 3

Applying the rigid facets subdivision algorithm to the PSL 4 (Z)-equivariant cell complex from [START_REF] Mathieu | On the integral homology of PSL 4 (Z) and other arithmetic groups[END_REF], extracting the 3-torsion subcomplex, and reducing it using the methods of [START_REF] Rahm | Accessing the cohomology of discrete groups above their virtual cohomological dimension[END_REF], we get the following graph of groups T , decorated with the groups stabilizing the cells that are the pre-images of the projection to the quotient space.

S 3

S 3 S 3 × S 3 S 3 × S 3 E 9 ⋊ C 2 C 3 S 3 E 9 ⋊ C 2
Here, E 9 ⋊ C 2 denotes the semi-direct product of the elementary abelian group of order 9 with C 2 , where C 2 acts by inversion (corresponding to the group [START_REF] Rubén | Equivariant K-homology for some Coxeter groups[END_REF][START_REF] Bui | Computing Bredon homology of groups[END_REF] in GAP's SmallGroups library). The machine computation provided the following system of morphisms among the above cell stabilizers. The E 9 ⋊ C 2 edge stabilizer admits an isomorphism of groups φ (not the identity, though) to the E 9 ⋊ C 2 vertex stabilizer and an inclusion into the S 3 × S 3 vertex stabilizer. Of the two S 3 edge stabilizers, one has maps diag(1, 1) and diag(1, 0) to the two S 3 × S 3 vertex stabilizers, and the other one has maps diag(1, -1) and diag(-1, -1) to the two S 3 × S 3 vertex stabilizers. The C 3 edge stabilizer admits an inclusion into the E 9 ⋊ C 2 vertex stabilizer, and an inclusion into the S 3 vertex stabilizer.

By the properties of torsion subcomplex reduction, the PSL 4 (Z)-equivariant cohomology of the 3-torsion subcomplex is isomorphic to the PSL 4 (Z)-equivariant cohomology of the above graph of groups T . Similarly, the Farrell-Tate cohomology of PSL 4 (Z) at the prime 3 is isomorphic to the Farrell-Tate cohomology of the above graph of groups. In the following, we evaluate the isotropy spectral sequence

E p,q 1 = σ∈Tp H q (Stab(σ); F 3 ) ⇒ H p+q (PSL 4 (Z); F 3 )
converging to Farrell-Tate cohomology. As we only consider a graph, the spectral sequence is concentrated in the two columns p = 0, 1. The differential d 1 is induced from the inclusions of subgroups, up to the sign coming from the choice of orientation of the graph. Since the spectral sequence is only concentrated in the first two columns, we will have E 2 = E ∞ . Since we are interested in field coefficients, there are no extension problems to solve at the E ∞ -page.

The relevant cohomology groups of the finite groups are:

• H • (C 3 ; F 3 ) ∼ = F 3 [x](a) with deg a = 1 and deg x = 2. • H • (S 3 ; F 3 ) ∼ = F 3 [y](b) with deg b = 3 and deg y = 4. • By the Künneth formula, H • (S 3 × S 3 ; F 3 ) ∼ = H • (S 3 ; F 3 ) ⊗2 .
• By the Hochschild-Serre spectral sequence,

H • (E 9 ⋊ C 2 ; F 3 ) ∼ = F 3 [x 1 , x 2 ](a 1 , a 2 ) C2 where deg x i = 2, deg a i = 1
and C 2 acts by multiplication with -1 on all the generators. To describe the d 1 -differential, it is enough to note that the restriction map associated to the inclusion C 3 ֒→ S 3 is the inclusion of C 2 -invariants. Now, for the evaluation of the spectral sequence, we first deal with the edges attached to the loop.

(1) The restriction map

φ ⊕ (Res S3 C3 • pr * 2 ) : H • (E 9 ⋊ C 2 ; F 3 ) → H • (E 9 ⋊ C 2 ; F 3 ) ⊕ H • (C 3 ; F 3
) is injective, and the cokernel is isomorphic to H • (C 3 ; F 3 ). Here φ denotes the isomorphism E 9 ⋊ C 2 ∼ = E 9 ⋊ C 2 appearing as stabilizer inclusion in the reduced torsion subcomplex.

(2) The inclusion of the dihedral vertex group into the cyclic edge group is an injection

Res S3 C3 : H • (S 3 ; F 3 ) ֒→ H • (C 3 ; F 3 )
given by the inclusion of the invariant elements for the C 2 -action by -1. Therefore, the cokernel is concentrated in degrees 1, 2 mod 4 (except for the degree 0). Therefore, we can reduce the E 1 -page of the spectral sequence as follows: from (1), we find that we can remove the two summands for E 9 ⋊ C 2 from the columns p = 0 and p = 1, respectively; but in turn, we have to replace the restriction map for E 9 ⋊ C 2 ֒→ S 3 × S 3 by a map from the cohomology of S 3 × S 3 to the cokernel of the restriction map in (2). However, since the latter is concentrated in degrees 1 and 2, the induced restriction map is trivial in the generating degrees 3 and 4, hence it is trivial. Therefore, the E 1 -page decomposes: one contribution comes from the cokernel of the restriction map H • (S 3 ; F 3 ) ֒→ H • (C 3 ; F 3 ), the other contribution comes from the loop connecting the two copies of S 3 × S 3 via the S 3 -edges.

The cohomology of the loop is computed as follows:

(3) The restriction maps diag(1, 1), diag(1, 0), diag(1, -1) and diag(-1, -1) :

F 3 [x 4 , y 4 ](a 3 , b 3 ) ∼ = H • (S 3 × S 3 ; F 3 ) → H • (S 3 ; F 3 ) ∼ = F 3 [z 4 ](c 3 )
are surjective. On the kernel of diag(1, 1), the restriction of diag(1, -1) is still surjective. On the kernel of diag(-1, -1), the restriction of diag(1, 0) is still surjective. Therefore, the cohomology of one edge of the loop is killed already by the restriction map from any one of the vertex groups. Number the S 3 × S 3 -vertices by 1 and 2, and the S 3 -edges by a and b. The restriction from the vertex group 1 to the edge a is surjective. Removing this part from the spectral sequence, the restriction from the vertex group 2 to the edge a is trivial, but we still have the restriction to the edge b. This kills the edge cohomology b, showing that the differential d 1 is surjective in the loop part of the E 1 -page. The kernel of the differential consists then exactly of two copies of the kernel of a restriction map Res S3 × S3 S3 .

Theorem 10. The Farrell-Tate cohomology of the group PSL 4 (Z) (with coefficients in F 3 ) in degrees 2 is given as follows:

H • (PSL 4 (Z); F 3 ) ∼ = ker Res S3 × S3 S3 ⊕2 ⊕ coker H •-1 (S 3 ; F 3 ) → H •-1 (C 3 ; F 3 ) .
This, in particular also computes the 3-torsion group cohomology of PSL 4 (Z) above the virtual cohomological dimension.

The cokernel of the d 1 -differential in degree 0 and hence the first cohomology H 1 (PSL 4 (Z); F 3 ) is of F 3 -rank 1, coming from the loop of the 3-torsion graph.

Remark 11. The kernel comes from the p = 0 column of the E 2 = E ∞ -page. The cokernel comes from the p = 1 column and consequently has a shift. The cup product on the kernel is the one induced from cohomology of S 3 × S 3 , and cup-product with classes in the cokernel is always 0.

Inspired by Grunewald, we consider the Hilbert-Poincaré series of the Farrell-Tate cohomology of PSL 4 (Z) with coefficients in F ℓ :

HP PSL4(Z) (T ; ℓ) := ∞ q=1 dim H q (PSL 4 (Z); F ℓ ) • T q .
Corollary 12. The Hilbert-Poincaré series of the 3-torsion Farrell-Tate cohomology of PSL 4 (Z) (for degrees 1) is then

HP PSL4(Z) (T ; 3) = T + 2(T 3 + T 4 + T 6 + T 7 ) (1 -T 4 ) 2 + T 2 + T 3 1 -T 4 .
Remark 13. Note that the above calculation describes the Farrell-Tate cohomology in all degrees, not just some small ones. Essentially, the computer calculation produces the reduced torsion subcomplex (which encodes the cohomology for all degrees). The spectral sequence is evaluated using the cup-product structure. Note that the finiteness results for group homology imply that the cup-product structure for both group and Farrell-Tate cohomology is finitely generated. Using suitable commutative algebra packages, such computations of the ring structure (and therefore additive computations for all cohomological degrees) could probably also be automated.

We make the following consideration on the compatibility of our result for Farrell-Tate cohomology with the result of Dutour-Ellis-Schürmann [START_REF] Mathieu | On the integral homology of PSL 4 (Z) and other arithmetic groups[END_REF] for group homology in low degrees. The isomorphism types computed in the latter article are to correspond as follows to the evaluation of our above Hilbert-Poincaré series in those degrees.

H q (PSL 4 (Z); Z) ∼ =              0, q = 1, (Z/2) 3 , q = 2, Z ⊕ (Z/4) 2 ⊕ (Z/3) 2 ⊕ Z/5, q = 3, (Z/2) 4 ⊕ Z/5, q = 4, (Z/2) 13 , q = 5, - dim H q (PSL 4 (Z); F 3 ) =             
1, q = 1, 1, q = 2, 3, q = 3, 2, q = 4, 0, q = 5.

For this to be consistent, the Farrell-Tate cohomology groups in degrees 1 and 2 need to vanish in group homology; so, these should be annihilated by differentials from the orbit space. We have evidence for this in degree 1, since the loop in the graph becomes contractible in the orbit space of the full locally symmetric space. In degree 3, one of the summands in H 3 (PSL 4 (Z); Z) is rationally non-trivial and must come from the orbit space. This means that only the submodule (Z/3) 2 can come from Farrell-Tate cohomology, and the third dimension that we observe in degree 3 Farrell-Tate cohomology must belong to the degree 2 stabilizer cohomology class that is annihilated by the above mentioned differentials from the orbit space.

From Theorem 10, we deduce that the degree 2 Farrell-Tate class can only come from

coker H •-1 (S 3 ; F 3 ) → H •-1 (C 3 ; F 3 ) .
Then, this class and its group homology counterpart sit at position p = 1, q = 1 in the respective equivariant spectral sequence, and hence the annihilating differential, emanating from the orbit space homology module Z ⊂ H 3 (PSL 4 (Z); Z) sitting at position p = 3, q = 0, must be of second degree.

In degrees 4 and 5, the dimensions already agree via the Universal Coefficient Theorem, so here we infer that the submodule (Z/3) 2 in degree 3 should actually come from Farrell-Tate cohomology, so it should be stabilizer cohomology that is not hit by higher degree differentials.

Homological 5-torsion in PSL 4 (Z)

We applied the rigid facets subdivision algorithm to the PSL 4 (Z)-equivariant cell complex of [START_REF] Mathieu | On the integral homology of PSL 4 (Z) and other arithmetic groups[END_REF], extracted the 5-torsion subcomplex, and reduced it using the methods of [START_REF] Rahm | Accessing the cohomology of discrete groups above their virtual cohomological dimension[END_REF] to the following graph T :

D 5 D 5
The d 1 -differential of the equivariant spectral sequence on T is zero, because the isomorphisms at edge end and edge origin cancel each other. Then the E 1 = E ∞ page is concentrated in the columns p = 0 and 1, with dimensions over F 5 being 1 in rows q congruent to 3 or 4 mod 4, and zero otherwise. This yields Proposition 14. We observe on Farrell cohomology: dim F5 H p+q (PSL 4 (Z);

F 5 ) =         
1, p + q ≡ 1 mod 4, 0, p + q ≡ 2 mod 4, 1, p + q ≡ 3 mod 4, 2, p + q ≡ 4 mod 4.

We check this result with a computation of H

] = O Q(ζ5) is a Dedekind ring.
We can therefore use Reiner's result [START_REF] Reiner | Integral representations of cyclic groups of prime order[END_REF] to determine conjugacy classes of C 5 -subgroups in GL 4 (Z). Since both Z and Z[ζ 5 ] have trivial class group, there is only one isomorphism class of Z[C 5 ]-module with nontrivial action and Z-rank 4. Hence, there is a unique conjugacy class of cyclic order 5 subgroup in GL 4 (Z). Since the center of GL 4 (Z) is of order 2, the same is true for PGL 4 (Z). Now there is a necessary modification to deal with the case SL 4 (Z), along the lines of the discussion in [START_REF] Rahm | On Farrell-Tate cohomology of SL 2 over S-integers[END_REF]. While conjugacy classes of C 5 -subgroups in GL 4 (Z) correspond to isomorphism classes of Z[C 5 ]-modules, the conjugacy classes of C 5 -subgroups in SL 4 (Z) correspond to such modules equipped with an additional orientation, i.e., a choice of isomorphism det M ∼ = The centralizer of this C 5 -subgroup is the group of norm-1 units of Z[ζ 5 ], which by Dirichlet's unit theorem is isomorphic to ker

Z[ζ 5 ] × → Z × ∼ = Z/10Z × Z.
As in [15, Section 5], the normalizer is an extension of the centralizer by an action of the stabilizer of the corresponding oriented module in the Galois group. We noted above that the Galois group Z/4Z exchanges the two orientations of the trivial module, hence the stabilizer is the subgroup Z/2Z ⊂ Z/4Z. The normalizer therefore is of the form (Z/10Z × Z) ⋊ Z/2Z. The action of Z/2Z on Z/10Z is by multiplication with -1 because the action of the Galois group is via the identification Z/4Z ∼ = Z/5Z × . The action of Z/2Z on Z is trivial: the full Galois group acts on Z via a surjective homomorphism Z/4Z → Z × ∼ = Z/2Z. The stabilizer of the oriented module in the Galois group lies in the kernel of the above action, as claimed. Therefore, the normalizer is in fact of the form D 10 ×Z.

Applying the formulas from [15, Section 3], the Farrell-Tate cohomology of the normalizer is of the form F 5 [a ±2 2 ](b 3 1 ) ⊕2 ⊕ F 5 [a ±2 2 ](b 3 1 ) ⊕2 -1 where the lower subscript -1 indicates a degree shift by -1. The Hilbert-Poincaré series for the positive degrees is T 3 +2T 4 +T 5 1-T 4 = T 3 (1+T ) 2

1-T 4 . The computations in [START_REF] Mathieu | On the integral homology of PSL 4 (Z) and other arithmetic groups[END_REF] show that the 5-torsion in integral homology of PSL 4 (Z) of dimension 1 in degrees 0, 3 mod 4 and trivial otherwise. By the universal coefficient theorem, this agrees with the above computation.

Necessity of Rigidity for Bredon homology

From a non-rigid cell complex, i.e., a cell complex where cell stabilizers do not necessarily fix the corresponding cell pointwise, one can compute classical group homology via the equivariant spectral sequence with coefficients in the orientation module. Such an orientation module, where elements of the stabilizer group act by multiplication with 1 or -1, depending on whether they preserve or reverse the orientation of the cell, cannot exist for Bredon homology. We make this precise in the following statement: Proposition 15. There is no module-wise variation of the Bredon module with coefficients in the complex representation ring and with respect to the system of finite subgroups such that Bredon homology can be computed from a non-rigid cell complex.

Proof. We provide a counterexample in order to contradict the possibility to compute the Bredon homology from an arbitrary non-rigid cell complex.

Consider the classical modular group PSL 2 (Z). A model for EPSL 2 (Z) is given by the modular tree [20]. There is a rigid cell complex structure T 1 on it, given as follows. By [20], the modular tree admits a strict fundamental domain for PSL 2 (Z), of the shape

Z/3Z Z/2Z
with vertex stabilizers as indicated and trivial edge stabilizer. We obtain the cell complex T 1 by tessellating the modular tree with the PSL 2 (Z)-images of this fundamental domain. Obviously, T 1 is rigid, and it yields the Bredon chain complex

0 → R C ( 1 ) d -→ R C (Z/2Z) ⊕ R C (Z/3Z) → 0.
The map d in the above Bredon chain complex is injective, and as R C (Z/nZ) ∼ = Z n , we read off H Fin 1 (EPSL 2 (Z); R C ) = 0, H Fin 0 (EPSL 2 (Z); R C ) ∼ = Z 4 . Now we equip the modular tree with an alternative equivariant cell structure T 2 , induced by the non-strict fundamental domain Z/3Z Z/3Z

where the (set-wise) edge stabilizer is Z/2Z, flipping the edge onto itself. It can be seen as a ramified double cover of the fundamental domain for T 1 discussed above. A system of representative cells for T 2 is given by the edge of double length, and one vertex of stabilizer type Z/3Z. This yields a chain complex 0 → R C (Z/2Z) → R C (Z/3Z) → 0, where the tilde could be any construction which takes the non-trivial Z/2Z-action on the edge of double length into account (similar to the coefficients in the orientation module for group homology computed from non-rigid cell complexes). But no matter how this construction is done, from R C (Z/3Z) ∼ = Z 3 , we can never reach H Fin 0 (EPSL 2 (Z); R C ) ∼ = Z 4 . Hence T 2 is our desired counterexample.

Remark 16. We could of course drop the condition "module-wise" in the above proposition, and investigate whether there is a reasonable construction which maps the representation ring to a complex of modules and yields a quasi-isomorphism from the total complex to the Bredon complex for the subdivided tree. But with such a construction, one would only superficially hide the fact that one needs to know how to subdivide in order to get the constructed complexes right. This means that it will not be practicable to compute Bredon homology with respect to the system of finite

  In BCS(X) : # Orbits | # Cells 11 | 513 90 | 3488 295 | 7904 368 | 7392 154 | 2464 present a new implementation of Torsion Subcomplex Reduction, which will overcome this problem using Discrete Morse Theory. 7. Example: Farrell-Tate cohomology of Sp 4 (Z) at odd primes

4 Z

 4 M ∼ = Z. The conjugacy class in GL 4 (Z) lifts to SL 4 (Z), and the corresponding module has two different choices of orientation. The Galois group Gal(Q(ζ 5 )/Q) ∼ = Z/4Z acts on the set of oriented modules. The action exchanges the orientations. Therefore, there is one conjugacy class of C 5 -subgroup in SL 4 (Z) stabilized by Z/2Z ֒→ Gal(Q(ζ 5 )/Q).

Table 2 .

 2 Time spent on a single processor for subdividing available cell complexes.Arithmetic groupSL 3 (Z) Γ 1 in SL 3 (Z) Γ 2 in SL 3 (Z) Sp 4 (Z) PSL 4 (Z)

	Rigid Facets Subdivision	8s	17s	26s	24s	69 minutes
	Virtually Simplicial Subdivision	9s	22s	33s	43s 111 minutes
	Hybrid Subdivision	9s	20s	31s	33s	33 minutes
	Barycentric Subdivision	13s	56s	104s	296s	?

The rigid facets subdivision algorithmIn this section, we discuss the rigid facets subdivision algorithm which rigidifies equivariant cell complexes. The core of the method is Algorithm 2, which is expected to run in reasonable time for input coming from cell complexes for arithmetic groups. The key fact which guarantees that rigid facets subdivision works, is Lemma 7 below.

Acknowledgements. We are grateful for support by Gabor Wiese's Fonds National de la Recherche Luxembourg grants (INTER/DFG/FNR/12/10/COMFGREP and AMFOR), which did facilitate meetings for this project via visits to Université du Luxembourg by the first author (thrice for one month) and the third author. The first author was funded by Vietnam National University -Ho Chi Minh City (VNU-HCM) under grant number C2018-18-02. We would like to thank Graham Ellis for having supported the development of the first implementation of our algorithms -the "Torsion Subcomplexes Subpackage" for his Homological Algebra Programming (HAP) package in GAP. Further thanks go to Sebastian Schönnenbeck [1,19] for having provided us cell complexes for congruence subgroups in SL 3 (Z), used for benchmarking purposes in this paper, and especially to Mathieu Dutour Sikirić for having provided the cell complexes for SL 3 (Z), Sp 4 (Z) and PSL 4 (Z) in HAP. This article is dedicated to the memory of Aled Ellis.

subgroups and coefficients in the complex representation ring without subdividing the cell complex under consideration to make it rigid.