SYMMETRIC AND ALTERNATING GROUPS IN ODD CHARACTERISTIC
OLIVIER BRUNAT AND JEAN-BAPTISTE GRAMAIN
Abstract. In this paper, following the methods of [8], we show that the double covering groups of the symmetric and alternating groups havep-basic sets for any odd primep.
1. Introduction
LetGbe a finite group andCbe a union of conjugacy classes ofG. For any class functionαon G, define resC(α) =αb to be the class function onGthat takes the same values asαonCand 0elsewhere. A C-basic set ofGis a subsetb of the set
Irr(G)of complex irreducible characters of Gsuch thatbb={χb|χ∈b}is aZ-basis of theZ-module generated byIrr(G) =c {χb|χ∈Irr(G)}.
In [9, §2.1], the authors develop a generalized modular theory ofGwith respect toC. In particular, there is a partition ofIrr(G)intoC-blocks with respect to this C-structure. The construction goes as follows. Given anyZ-basisβ of theZ-module generated by Irr(G)c , two characters χ, χ0 ∈Irr(G) are calledβ-linked if χb andχb0 have a common factor when decomposed in the basisβ. The transitive closure of
β-linking gives a partition of Irr(G)into equivalence classes called C-blocks. Note that theC-blocks we obtain depend on the choice ofβ, but thatβ doesn’t need to be a basic set forG. Furthermore, by [9, Prop 2.14], there always exists a choice of basisβ such that theC-blocks coincide with the generalized blocks constructed by Külshammer, Olsson and Robinson in [23, §1], and defined by orthogonality across C. Throughout the paper, we will always assume that this specific choice of basis is made.
We can then analogously define the notion ofC-basic set of anyC-blockB. Note that, ifbis aC-basic set ofGandB is anyC-block ofG, thenb∩Bis aC-basic set ofB. Conversely, given aC-basic set for eachC-block ofG, one recovers aC-basic set ofG.
Letpbe a prime. WhenCis the set ofp-regular elements ofG, that is, elements whose order is prime top, theC-modular theory ofGis the classicalp-modular char-acter theory, and the notion ofC-basic set coincides with that ofp-basic set. One of the main challenges of Brauer’s theory is the determination of thep-decomposition matrix ofG, of which a first approximation can be obtained when one has ap-basic set ofG.
2010Mathematics Subject Classification. Primary 20C30, 20C15; Secondary 20C20. The second author was supported by the EPSRC grantCombinatorial Representation Theory EP/M019292/1.
Corresponding Author : Olivier Brunat.
While it is unclear how to exhibit ap-basic set ofGin general, Hiss conjectured that all finite groups do havep-basic sets. This conjecture is true in the following cases: for p-soluble groups [11, X.2.1], for certain finite reductive groups under various assumptions ([17], [12], [15], [13], [14], [22], [10], [6]), and for symmetric and alternating groups ([21], [8], [7], [2]).
The covering groups of the symmetric groups have a 2-basic set (see [4, Thm 5.2]). Forpodd, attempts have been made in [3] and [1] to generalize the method of James and Kerber, but these methods fail forp≥7. In this paper, we prove the following.
Theorem 1.1. The covering groups of the symmetric and alternating groups have p-basic sets for any odd primep.
In [9] (see also [23]), the authors gave a notion of perfect isometries, which generalizes toC the perfect isometries introduced by Broué in [5]. Such a perfect isometry I furthermore satisfies I(χ) =b I(χ)d for allχ ∈ B, whence in particular sends a basic set to a basic set.
To prove Theorem 1.1, we follow the strategy of [8]. In Section 3, we present groupsGeandHe and unions of conjugacy classesCofGeandDofHe, and construct aC-basic set ofGe (see Theorem 3.3) and aD-basic set ofHe (see Theorem 3.4).
In Section 4, we adapt the perfect isometry constructed by Livesey in [24], where he proves Broué’s Abelian Defect Perfect Isometry Conjecture, and we apply the methods and results of [9]. This produces, for anyp-blockB of a double covering group of the symmetric or alternating groups, a perfect isometry between B and a C-block of Ge or a D-block ofHe, from which Theorem 1.1 is deduced. Finally, Section 5 is devoted to some further remarks. In particular, we recover some results of Olsson on the Brauer characters inp-blocks of covering groups (podd), and their number.
2. Background
Letn be a positive integer. We writeSne for the double covering group of the symmetric groupSn defined by
e Sn =
z, ti,1≤i≤n−1 |z2= 1, t2i =z,(titi+1)3=z,(titj)2=z(|i−j| ≥2). The groupSne and its representation theory were first studied by I. Schur in [32], and, unless otherwise specified, we always refer to [32] for details or proofs. Note thatSnhas another, non-isomorphic, double covering group, usually denoted bySn. However, from the fact thatb Sne has ap-basic set, we will deduce thatSnb does as well (see Remark 4.10).
We recall that we have the following exact sequence
1→ hzi →Sen→Sn→1.
We denote byθ:Sne →Sn the natural projection. Note that for everys∈Sn, we have θ−1(s) = {
e
s, zse}, where es∈ Sne is such that θ(es) =s. Whenever G is a subgroup of Sn, we set Ge = θ−1(G). Any irreducible (complex) character of Ge withz in its kernel is simply lifted from an irreducible character of G. Any other irreducible characterξofGeis called aspin character, and it satisfiesξ(z) =−ξ(1). We denote bySI(G)e the set of irreducible spin characters ofGe. Defineε= sgn◦θ,
where sgnis the sign character ofSn. In particular, ifGis the alternating group An, thenAne = ker(ε).
Note that the restriction ofεtoGe (again denoted byε) is a linear (irreducible) character ofGe, and for any spin characterξofGe,ε⊗ξis a spin character (because
ε⊗χ(z) = −ε⊗χ(1)). A spin character ξ is said to be self-associate if εξ = ξ. Otherwise,ξandεξare called associate characters.
If H1 and H2 are subgroups of Sn acting non-trivially on disjoint subsets of
{1, . . . , n} and H = H1×H2 ≤ Sn, then Schur defined in [32] (see also [18])
thetwisted central product denoted by׈ and such thatHe =He1׈He2≤Sn. Schure proved that there is a surjective map⊗ˆ : SI(He1)×SI(He2)→SI(He). Schur explicitly described when two pairs of spin characters have the same image χ⊗ˆψ under this map, and how to detect whether χ⊗ˆψ is self-associate or not. Furthermore, we derive from case 1 of §2 and the proof of Theorem 2.4 of [18] that
Proposition 2.1. Let He1 and He2 be as above and not contained in the kernel of
ε. Let χ1 and χ2 be irreducible spin characters of He1 and He2. Let τ1 ∈ He1 and
τ2∈He2 be such thatε(τ1) =ε(τ2) = 1. Then
(1) Ifχ1 (or χ2) is self-associate, then
(χ1⊗ˆχ2)(τ1, τ2) =χ1(τ1)χ2(τ2).
(2) Ifχ1 andχ2 are non-self-associate, then
(χ1⊗ˆχ2)(τ1, τ2) = 2χ1(τ1)χ2(τ2).
Let Dn be the set of bar partitions of n, i.e. the partitions of n with distinct parts. Denote byD+
n (respectivelyD−n) the subset ofDnconsisting of all partitions
π∈ Dn such that the number of even parts ofπis even (respectively odd). In [32], Schur proved that the spin characters ofSne are, up to association, labeled byDn. More precisely, he showed that everyλ∈ D+
n indexes a self-associate spin character
ξλ, and everyλ∈ D−
n a pair(ξ
+
λ, ξ
−
λ)of associate spin characters.
The irreducible spin characters of Ane are also labeled by the bar partitions of
n. Ifλ∈ D+ n, thenRes e Sn e An
(ξλ) =ζλ++ζλ− with ζλ+, ζλ− ∈SI(Ane ). Ifλ∈ D−n, then
ResSen e An (ξλ+) = ResSen e An (ξλ−) =ζλ∈SI(Aen).
For any bar partition λ= (λ1 > · · · > λk >0) of n, we set, as in the case of partitions,|λ|=P
λiand we define thelength`(λ)ofλby`(λ) =k. Furthermore, we setσ(λ) = (−1)|λ|−`(λ). With this notation, we then haveλ∈ Dσn(λ)(see e.g. [31, p. 45]).
Letpbe an odd integer. To anyλ∈ Dn, one can associate in a canonical way its
p-coreλ(p)and itsp-quotientλ(p); see [31, p. 28]. Thep-coreλ(p)is a bar partition
ofn−wp, wherewis a non-negative integer called thep-weight ofλ. Thep-quotient has the formλ(p)= (λ0, λ1, . . . , λ(p−1)/2), whereλ0is a bar partition, theλi’s are partitions for1≤i≤(p−1)/2, and the sizes of the λi’s (0≤i≤(p−1)/2) add up to w. If we write, in analogy with bar partitions, σ(λ(p)) = (−1)w−`(λ0)
, then we obtain that
3. Some basic sets
Let w be a positive integer and p be an odd prime. In this section, we will focus on a groupGe which, whenw < p, is isomorphic to the normalizer of a Sylow
p-subgroup ofSpw. We will recall the description of its irreducible characters, ande describe a unionCof conjugacy classes and aC-basic set forGe.
SetK=Z/pZ o Z/(p−1)ZandG=KoSw. Recall thatGis a natural subgroup of Spw and that G=NoSw, where N =Kw. Now, let Q= (Z/pZ)w/ N and
L=Z/(p−1)ZoSw. Then G=QoL. Note that |Qe|= 2pw. Thus, by Sylow’s theorems,Qehas a Sylowp-subgroupP of index2, whence normal and unique inQe. Sincehziis normal inQeandhzi ∩P is trivial (aspis odd), we see thatQe=hzi ×P and thatP 'Q. Now take anyg ∈Ge. Then, sinceQ /e Ge, the group gP g−1 is a Sylowp-subgroup ofQe. Hence,gP g−1=P andP is normal inGe. Furthermore, as
Q∩L= 1, we haveP∩Le= 1(becauseP∩Le=P∩(Qe∩L) =e P∩ hzi= 1). Hence
(2) Ge=NeSwe =PoL.e
The irreducible spin characters ofGewere constructed by Michler and Olsson in [25] using Clifford theory (see for example [20, Chap 6]) ofGewith respect to the normal subgroupNe. The parametrization they obtain is as follows.
Proposition 3.1. The irreducible spin characters ofGe can be labeled explicitly by
the p-quotients of w. Ifq= (q0, q1, . . . , q(p−1)/2)is ap-quotient of w, thenq labels two associate charactersψ+
q andψ−q if and only ifσ(q) =−1; otherwise, qlabels a
unique self-associate character ψq.
For the second statement, an explanation can be found for example in [16, p. 422]. Proposition 3.2. The spin characters ofGewhich havePin their kernel are exactly
those labeled byp-quotients of the formq= (∅, q1, . . . , q(p−1)/2).
Proof. Take any ψ ∈ Irr(G)e . Since P ≤ Ne, to study the restriction of ψ to P, we will look at the restriction ofψto Ne. On the other hand, the restriction toNe is given by Clifford’s theorem [20, Thm 6.2] which goes as follows. There exist a uniqueGe-orbitΩψ of irreducible characters ofNe and a unique positive integereψ such that (3) ResGe e N(ψ) =eψ X χ∈Ωψ χ.
It follows from (3), the fact thatψis a spin character and that allχ∈Ωψtake the same value on z thatΩψ consists only of spin characters. We will thus now recall the description given in [25] of theGe-orbits of spin characters of Ne.
The irreducible spin characters of Ne are of the form χ1⊗ · · ·ˆ ⊗ˆχw, where χi ∈
SI(K)e for1 ≤i≤w. As above, we have Ke =hτioθ−1(Z/(p−1)Z), where τ is an element or orderp. In [25], Michler and Olsson show thatKe has exactlypspin characters: a unique self-associate η0, of degree p−1 and with η0(τ) =−1, and
(p−1)/2 pairs of associate linear characters η+1, η−1, . . . , η(+p−1)/2, η(−p−1)/2, which are exactly those havinghτiin their kernel.
For any decompositiont= (t0, t1, . . . , t(p−1)/2)ofw, we set (4) θt=η0⊗ · · ·ˆ ⊗ˆη0 | {z } t0 factors ˆ ⊗η+1⊗ · · ·ˆ ⊗ˆη+ 1 | {z } t1 factors ˆ ⊗ · · ·⊗ˆη+ (p−1)/2⊗ · · ·ˆ ⊗ˆη + (p−1)/2 | {z } t(p−1)/2 factors ∈SI(Ne).
Now supposeψis labeled by thep-quotientq= (q0, . . . , q(p−1)/2)ofw. From [25, Prop 3.12] and the construction ofψ, we get that, if|q0|>1, thenθ(|q0|, ...,|q(p−1)/2|)
is a representative forΩψ= Ωεψ (even ifψ6=εψ). If|q0| ≤1 andw− |q0|is odd, thenψ6=εψand representatives forΩψ andΩεψare given byθ(|q0|, ...,|q(p−1)/2|)and
εθ(|q0|, ...,|q(p−1)/2|) (not necessarily in this order). In all cases, we writeθψ for the representative ofΩψ.
Note that Q = hu1i × · · · × huwi, where ui has order p for 1 ≤ i ≤ w. For
1≤i≤w, letτi be the unique element of orderpin θ−1({ui}). Then
P =hτ1i × · · · × hτwi.
In particular,P ≤Apw.e
Now, by (3), we see that, for any(x1, . . . , xw)∈P, we have
ψ(x1, . . . , xw) =eψ X χ∈Ωψ
χ(x1, . . . , xw).
In particular, we haveψ(1) =eψ Pχ∈Ωψχ(1) =eψθψ(1)|Ωψ|. Furthermore, iterat-ing Proposition 2.1 shows that, for anyχ∈Ωψ, there is a non-negative integerαχ such that χ(1) = 2αχ(p−1)|q0| and χ(τ
1, . . . , τw) = 2αχ(−1)|q
0|
. Since χ(1) does not depend on the choice of χin Ωψ, neither doesαχ which we thus denote byα. In particular, we have ψ(1) =eψ2α(p−1)|q 0| |Ωψ| and ψ(τ1· · ·τw) =eψ2α(−1)|q 0| |Ωψ|. It follows that, if q0 6=∅, then ψ(τ
1· · ·τw)6=ψ(1), whence P is not contained in
ker(ψ). Ifq0=∅, then a similar computation shows that, for any(x1, . . . , xw)∈P, we haveψ(x1, . . . , xw) =eψ2α|Ωψ|=ψ(1), as required. We defineCto be the union of conjugacy classes ofGewhich have a representative inLe.
Theorem 3.3. The set of irreducible spin characters of Ge is a union of C-blocks,
with C-basic set the set B of spin characters labeled by p-quotients of the form q= (∅, q1, . . . , q(p−1)/2).
Proof. Letχbe a non-spin character ofGeandψbe a spin character ofGe. For any
g∈Ge, one hasχ(g) =χ(zg)andψ(g) =−ψ(zg). Also, since z∈Le, we have that C=zC. Thus, writing, for anyA⊆Ge, hψ, χiA=|1
e G| P x∈Aψ(x)χ(x), we have 2|Ge|hψ, χiC =|Ge|(hψ, χiC +hψ, χizC) = X g∈C (ψ(g) +ψ(zg))χ(g) = 0.
This proves that spin and non-spin characters are orthogonal across C, so that the set of spin characters is a union ofC-blocks, as defined by Külshammer, Olsson and Robinson in [23, §1]. By [9, Prop 2.14], this in turn implies that it is a union ofC-blocks, as defined in Section 1.
By Lemma 4.1 and Proposition 4.2 of [8], the set of irreducible characters ofGe withP in their kernel is aC-basic set ofGe. The intersection of this basic set with
the set of spin characters ofGe, which is given by Proposition 3.2, is thus aC-basic
set for the set of spin characters ofGe.
LetH =G∩ Apw. By applying Clifford theory to H /e Ge, the irreducible spin characters of He are again labeled by the p-quotients of w. Any p-quotient q of
w labels two characters ϕ+q and ϕq− of He when σ(q) = 1, and one character ϕq otherwise. SetD=C ∩He.
Theorem 3.4. The set of irreducible spin characters ofHe is a union of D-blocks,
with D-basic set the set B0 of spin characters labeled by p-quotients of the form q= (∅, q1, . . . , q(p−1)/2).
Proof. Sincez lies inD, we conclude as in Theorem 3.3 that the set of irreducible spin characters ofHe is a union ofD-blocks. Furthermore, a simple counting argu-ment shows that
e
H =Po(Le∩Aepw).
Let q = (q0, q1, . . . , q(p−1)/2) be a p-quotient of w. If σ(q) = −1, then ϕ
q =
ResGe e H(ψ
+
q )hasP in its kernel if and only ifψ+q does. Ifσ(q) = 1, thenRes e G e H(ψq) =
ϕ+
q +ϕ−q, withϕ+q(1) =ϕ−q(1) =ψq(1)/2. Let x∈P. We have ψq(x) =ψq(1)if and only ifϕ+
q(x) +ϕ−q(x) =ψq(1); since, by [20, Lemma 2.15],|ϕ+q(x)| ≤ψq(1)/2 and|ϕ−q(x)| ≤ψq(1)/2, this holds if and only ifϕ+q(x) =ϕ−q(x) =ψq(1)/2.
This proves that any spin character ofHe labeled byqhasP in its kernel if and only ifq0=∅, and we can conclude as in the proof of Theorem 3.3.
Remark 3.5. The same arguments as in the proofs of Theorems 3.3 and 3.4 show that the set of characters ofGwithQin their kernel is aθ(C)-basic set ofG(which is the same as that found in [8]), and that the set of characters ofH withQin their kernel is aθ(D)-basic set ofH.
4. Perfect isometries and basic sets for the covering groups We start by recalling the definitions of perfect isometry and of Broué isometry, and we prove some of their properties which will be useful later.
Let G and G0 be two finite groups, and C and C0 unions of conjugacy classes
of Gand G0 respectively. Let B (resp. B0) be a union ofC-blocks ofG (resp. of C0-blocks ofG0); see [9, p. 4]. An isometryI :
CB →CB0 such thatI(ZB) =ZB0 is a perfect isometry if
(5) I◦resC = resC0◦I.
Remark 4.1. Note the change of terminology. What we call “perfect isometry” in this paper is what is referred to as “generalized perfect isometry” in [9, §2.4]. Remark 4.2. IfI:CB→CB0 andJ :CB0→CB00are perfect isometries, then it follows easily from (5) thatJ◦I:CB→CB00 is a perfect isometry. It also follows from (5) thatI−1:
CB0→CB is a perfect isometry.
ToI, we associateIb∈C(B×B0)as in [9, §2.3] which satisfies
(6) Ib=
X χ∈B
Recall that (see [9, Eq (16)]), for anyψ∈CIrr(B)andy∈G0, we have (7) I(ψ)(y) = 1 |G| X x∈G b I(x, y)ψ(x).
Using (6) and (7), an easy computation shows that, ifB00is a union ofC00-blocks
ofG00andJ :CB0 →CB00is a perfect isometry, then for any (x, z)∈G×G00
(8) J[◦I(x, z) = 1 |G0| X y∈G0 b I(x, y)J(y, z).b
Ifpis a prime and(K,R, k)is a splittingp-modular system forGandG0, and if CandC0 are the sets ofp-regular elements ofGandG0, thenI is a Broué isometry
if the following two properties are satisfied.
(i) For every(x, x0)∈G×G0,I(x, xb 0)lies in|CG(x)|R ∩ |CG0(x0)|R. (ii) IfI(x, xb 0)6= 0, thenxandx0 are either bothp-regular or both p-singular. Note that, in this case, I is also a perfect isometry in the above sense since (ii) implies (5) by [9, Prop. 2.15].
Lemma 4.3. If I : CB → CB0 and J : CB0 → CB00 are Broué isometries, then
J◦I:CB→CB00 is a Broué isometry.
Proof. Take any x∈Gand z ∈G00. First note that y 7→
b
I(x, y)Jb(y, z) is a class function ofG0. Thus, by (8), [ J◦I(x, z) |CG(x)| = X y∈G0/∼ b I(x, y) |CG(x)| b J(y, z) |CG0(y)| ∈ R.
A similar argument shows thatJ[◦I(x, z)∈ |CG(z)| R, whenceJ[◦I satisfies (i). Now supposeJ[◦I(x, z)6= 0. Then, by (8), there existsy∈G0such thatI(x, y)b 6= 0
andJb(y, z)6= 0; (ii) follows.
Remark 4.4. Given a Broué isometry I, a formula for Id−1 can be found in [9, Remark 2.12]. It easily implies thatI−1is a Broué isometry.
Theorem 4.5. Letpbe an odd prime andBbe ap-block ofSne such that{ξλ+, ξλ−} ⊆
B for some λ ∈ D−
n. Let J : CB → CB be the isometry given by J(ξ
+
λ) = ξ
−
λ,
J(ξλ−) =ξλ+, andJ(ξ) =ξ ifξ /∈ {ξ+λ, ξλ−}. Then J is a Broué isometry.
Proof. Recall (see [32]) that a set of conjugacy classes of Sne (the split classes) outside which all spin characters vanish can be labeled by the elements ofOn∪ D−
n, whereOn is the set of partitions ofn in odd parts. Eachπ∈ On∪ D−
n labels two conjugacy classes with representativestπ andztπ such that the following hold. For allπ∈ On,ξ+λ(tπ) =ξλ−(tπ)andξλ+(ztπ) =ξλ−(ztπ). For allπ∈ D−n,
ξ+λ(tπ) =ξ−λ(ztπ) =−ξλ+(ztπ) =−ξ−λ(tπ) =δπλi(n−k+1)/2 p
zλ/2, whereλ= (λ1>· · ·> λk >0) andzλ=λ1· · ·λk (see [32]).
Note that, forπ∈ On∪ D−
n, we have{tπ, ztπ}=θ−1(σπ), whereθ: Sne −→Sn is the natural projection, andσπ has cycle typeπ. If π= (1m1,2m2, . . .), we thus have|C e Sn(tπ)|=|CSen(ztπ)|= 2|CSn(σπ)|= 2Πj≥1j mj(m j!). In particular, since λ= (λ1>· · ·> λk>0), we have|CSe n(tλ)|=|CSen(ztλ)|= 2λ1· · ·λk= 2zλ.
Denote by I the identity onCB. By [9, Theorem 4.21],I is a Broué isometry, whence satisfies properties (i) and (ii).
First note thatJb(x, x0)vanishes wheneverxorx0 doesn’t belong to a split class of Sen. Properties (i) and (ii) are thus immediate in this case, and it is enough to considerx, x0 ∈ {tπ, ztπ|π ∈ On ∪ D−n}. Also, because of the definition of J and the description of the values ofξλ± given above, we see that, wheneverxorx0
belongs to{tπ, ztπ|π∈ On∪ Dn−\λ}, we haveJb(x, x0) =I(x, xb 0). Hence properties (i) and (ii) are also true forJ in this case.
It is therefore enough to study properties (i) and (ii) for J in the case where
x, x0 ∈ {tλ, ztλ}. Note that, in this case, (ii) is automatically satisfied sincetλand
zνtλ are bothp-regular or bothp-singular. It only remains to study property (i) in this case.
We will in fact consider Jb(x, x0)−I(x, xb 0), where x, x0 ∈ {tλ, ztλ}, and show that this lies in|CG(x)|R ∩ |CG0(x0)|R. SinceIdoes satisfy (i), this will show that
J also does. Takeν, ν0∈ {0, 1}. Then, by (6), b J(zνtλ, zν 0 tλ)−I(zb νtλ, zν 0 tλ) = ξ+λ(zνt λ)ξ−λ(z ν0 tλ) +ξλ−(zνtλ)ξ+λ(z ν0 tλ) −ξ+λ(zνt λ)ξ+λ(z ν0t λ) +ξλ−(zνtλ)ξλ−(z ν0t λ) .
If(n−k+ 1)/2is odd, then, by the above description,ξ±λ(zνt
λ), ξλ±(z ν0t λ)∈iR, and we haveξλ+(zνt λ) =ξλ−(z νt λ)andξλ−(zνtλ) =ξλ+(z νt λ). We thus obtain b J(zνtλ, zν 0 tλ)−I(zb ν tλ, zν 0 tλ) = ξλ−(zνtλ)ξλ−(z ν0 tλ) +ξ+λ(z ν tλ)ξ+λ(z ν0 tλ) −ξ−λ(zνtλ)ξλ+(z ν0t λ) +ξλ+(z νt λ)ξλ−(z ν0t λ) = ξλ+(zνtλ)−ξ−λ(z νt λ) ξλ+(zν0tλ)−ξ−λ(z ν0t λ) = 2ξλ+(zνtλ)2ξ+λ(z ν0t λ) =±4pzλ/2 2 =±2zλ. Since2zλ=|CSen(zνtλ)|=|CSen(zν 0
tλ)|, this shows that, in this case, (i) holds for b
J (andx=zνt
λ,x0=zν 0
tλ).
If, on the other hand,(n−k+ 1)/2 is even, thenξλ±(zνtλ), ξλ±(z ν0t λ)∈R, and we haveξλ+(zνt λ) =ξλ+(z νt λ)andξλ−(zνtλ) =ξλ−(z νt λ).
We thus obtain b J(zνtλ, zν 0 tλ)−I(zb νtλ, zν 0 tλ) = ξλ+(zνtλ)ξλ−(zν 0 tλ) +ξ−λ(zνtλ)ξ+λ(zν 0 tλ) −ξ+λ(zνtλ)ξλ+(zν 0 tλ) +ξλ−(zνtλ)ξλ−(zν 0 tλ) = ξλ+(zνtλ)−ξ−λ(z νt λ) ξλ−(zν0tλ)−ξ+λ(z ν0t λ) = 2ξλ+(zνtλ)(−2)ξ+λ(z ν0t λ) =±4pzλ/2 2 =±2zλ. Since 2zλ =|CSe n(z νt λ)|=|CSe n(z ν0t
λ)|, this shows that, in this case as well, (i) holds forJb(andx=zνtλ,x0=zν
0
tλ). This completes the proof. From now on, we fix an integern≥2and an odd primep. IfBis anyp-block of e
Sn, thenB contains either no or only spin characters (as can be seen by evaluating the central characters on the central elementz; see also [31]). In the former case,
B coincides with a p-block of Sn; in the latter, we say that B is a spin block. Similarly, any p-block ofAne contains either no spin character, and coincides with a p-block ofAn, or only spin characters, and is then called a spin block. The spin blocks ofSne andAne are described by the following:
Theorem 4.6 (Morris [27], Humphreys [19]). Let χ andψ be two spin characters ofSen, or two spin characters ofAen, labeled by bar partitions λandµrespectively.
Thenχis ofp-defect 0 (and thus alone in itsp-block) if and only ifλis ap-core. If λis not ap-core, thenχandψbelong to the samep-block if and only ifλ(p)=µ(p).
One can therefore define thep-core of a spin block B and its p-weight, as well as itssign σ(B) =σ(λ(p))(for any bar partitionλlabeling some characterχ∈B).
Theorem 4.7. Let B be any spinp-block ofSen of p-weight w >0. Let G,e He,C
andDbe as in §3, and letδp(λ)be the relative signof a bar partitionλintroduced
by Morris and Olsson in[28].
(i) Ifσ(B) = 1, thenI:CB→CSI(G)e given by
I(ξλ) =δp(λ)(−1)|λ 0| ψλ(p) if σ(λ) = 1, I(ξλ±) =δp(λ)(−1)|λ 0| ψ±λ(p) if σ(λ) =−1,
is a perfect isometry with respect top-regular elements ofSen andC.
(ii) Ifσ(B) =−1, thenI:CB →CSI(He)given by
I(ξλ) =δp(λ)(−1)|λ 0| ϕλ(p) if σ(λ) =−1, I(ξλ±) =δp(λ)(−1)|λ 0| ϕ±λ(p) if σ(λ) = 1,
is a perfect isometry with respect top-regular elements ofSen andD.
Proof. (i) We use the same convention as [24, p. 796] to label the cyclic structure of elements of G. In particular, if (π0, . . . , πp−1)is the cyclic structure associated
product (see [21, 4.2.1]) of orderp. Note that, with this convention,θ(C)is the set of elements with cyclic structure(π0, . . . , πp−1)such thatπ0=∅.
Let B0 be the principal spin p-block ofSpwe (that is, the spin p-block of Spwe corresponding to the emptyp-core), and consider the isometryI0:CB0→CSI(G)e given by I0(ξµ) =δp(µ)(−1)w(p 2−1)/8+w+|µ0| ψµ(p) ifσ(µ) = 1, I0(ξµ±) =δp(µ)(−1)w(p 2−1)/8+w+|µ0| ψηp(µ) µ(p) ifσ(µ) =−1, where ηp(µ) = ( ±δp(µ)(−1)w+ p−1 4 (|µ 0|−`(µ0)) ifσ(µ0) = 1, ±δp(µ)(−1)w+ p−1 4 (|µ 0|−`(µ0)+1) ifσ(µ0) =−1.
We first show thatI0is a perfect isometry with respect top-regular elements of
e
Sn andC. By [9, Prop. 2.15], all we have to prove is thatIb0(x, x0) = 0wheneverx
is ap-regular element of Sne andx0 ∈ C, or/ xisp-singular andx0 ∈ C. In order to prove this, we follow the strategy of the proof of [24, Thm 10.1]. First, we introduce the setCof elementst∈Sne such thatθ(t)has no cycles of length an odd multiple of p, as in [9, Eq. (60)], and the set C0 of elements t0 ∈
e
G such that the cyclic structure ofθ(t0)is (π
0, . . . , πp−1), with π0 having only even parts, as in [24, §10].
In the proof of [24, Thm 10.1], Livesey shows that I0 is a perfect isometry with
respect toC and C0. Note that, although Livesey is assuming in [24] thatw < p, this part of the proof runs through fine for generalw. Hence [9, Prop. 2.15] implies that Ib0(x, x0) = 0 wheneverx∈ C and x0 ∈/ C0, or x /∈ C and x0 ∈ C. To prove
that I0 is a perfect isometry with respect to the set of p-regular elements of Sne andC, it remains to show thatIb0(x, x0) = 0for any p-regular elementx∈Sne and
x0 ∈ C0\C, and for any p-singular element x∈C and x0 ∈ C. This last property is in fact already proved in the proof of [24, Thm 10.1]. Indeed, to show that I0
satisfies Property (ii) of a Broué isometry when p > w, Livesey uses repeatedly the sole argument that the cycle structure ofx0 is(∅, π1, . . . , πp−1)(which, by [24,
Lemma 7.2], coincides, whenp > w, with the fact thatx0 isp-regular). HenceI0 is
a perfect isometry with respect top-regular elements ofSne andC, as claimed. Using the fact that, if J is perfect isometry, then so is −J, we deduce that
I1 = (−1)w(p
2−1)/8+w
I0 is again a perfect isometry with respect to p-regular
el-ements of Sne and C. By [9, Thm 4.21], there exists a perfect isometry I2 with
respect to p-regular elements between B0 and B that associates characters of
B0 and B labeled by bar partitions with the same p-quotient. More precisely,
if λ(p) = µ(p), then I
2(ξµ) = δp(µ)δp(λ)ξλ if σ(µ) = 1, and I2({ξµ+, ξµ−}) = {δp(µ)δp(λ)ξλ+, δp(µ)δp(λ)ξλ−} ifσ(µ) =−1. Note that, sinceσ(B) =σ(B0) = 1, if
λ(p)=µ(p), thenσ(λ) =σ(µ).
It follows from Remark 4.2 thatI1−1 is a perfect isometry, and that the compo-sitionI2◦I1−1: SI(G)e →CB is a perfect isometry with respect toC andp-regular elements of Sen. If σ(λ) = 1, then I2◦I1−1(ψλ(p)) = δp(λ)(−1)|λ 0| ξλ, while, if σ(λ) =−1, then I2◦I1−1({ψ + λ(p), ψ − λ(p)}) ={δp(λ)(−1) |λ0|ξ+ λ, δp(λ)(−1) |λ0|ξ− λ}. Denote by Λthe set of partitions λsuch thatσ(λ) =−1 and I2◦I1−1(ψ
+
λ(p)) =
δp(λ)(−1)|λ
0|
ξλ−. Then, for eachλ∈Λ, we can define a Broué isometryIλ:CB−→ CB as in Theorem 4.5. Composing all the Iλ’s (λ ∈ Λ) gives a perfect isometry
I3:CB →CB with respect top-regular elements (by Remark 4.2), which is such that I3◦I2◦I1−1: SI(G)e →CB is given byI3◦I2◦I1−1(ψλ(p)) =δp(λ)(−1)|λ
0| ξλ if σ(λ) = 1 and I3◦I2◦I1−1(ψ ± λ(p)) = δp(λ)(−1) |λ0|ξ± λ if σ(λ) = −1. And, by Remark 4.2, I3◦I2◦I1−1 is a perfect isometry with respect to C and p-regular
elements ofSne . The result now follows from the fact that (9) I= (I3◦I2◦I1−1)
−1.
(ii) LetB0∗ be the principal spin p-block of Apw, and define the isometrye IA : CB0∗→CSI(H)e IA(ζλ) =δp(λ)(−1)|λ 0| ϕλ(p) ifσ(λ) =−1, IA(ζλ±) =δp(λ)(−1)|λ 0| ϕ±λ(p) ifσ(λ) = 1.
For all x ∈ Ane and x0 ∈ He, we set F(x, x0) = IbA(x, x0)− 12Jb(x, x0), where
J = (I3◦I2◦I1−1)−1:CB0 →CSI(G)e is the perfect isometry with respect to the set of p-regular elements of Sne and C given in Equation (9). For all x∈Ane and
x0∈He, we have F(x, x0) =A(x, x0) +B(x, x0), where A(x, x0) = X λ∈D−n δp(λ)(−1)|λ 0| ζλ(x)ϕλ(p)(x 0 )−1 2 ξ+ λ(x)ψ + λ(p)(x 0 ) +ξ−λ(x)ψ− λ(p)(x 0 ) and B(x, x0) = X λ∈Dn+ δp(λ)(−1)|λ 0| ζ+ λ(x)ϕ + λ(p)(x 0 ) +ζλ−(x)ϕ− λ(p)(x 0 )−1 2ξλ(x)ψλ(p)(x 0 ) .
Furthermore, for any λ ∈ D−
n and x, x0 as above, ξ + λ(x) = ξ − λ(x) = ζλ(x) and ψλ+(p)(x 0) = ψ− λ(p)(x 0) = ϕ λ(p)(x0). Hence, ξλ+(x)ψ+λ(p)(x 0) +ξ− λ(x)ψ − λ(p)(x 0) = 2ζλ(x)ϕλ(p)(x0), and A(x, x0) = 0.
Suppose now that λ∈ D+
n and that the cyclic structure of xis not of type λ. Then by [9, Equation (51)]ζλ+(x) =ζλ−(x) = 21ξλ(x), and for any x0∈He,
ζλ+(x)ϕ+λ(p)(x 0) +ζ− λ(x)ϕ − λ(p)(x 0) = 1 2ξλ(x)(ϕ + λ(p)(x 0) +ϕ− λ(p)(x 0)) = 1 2ξλ(x)ψλ(p)(x 0).
Thus,B(x, x0) = 0except ifxhas cycle typeλfor someλ∈ D+
n. In this case, for allx0 ∈ e H, B(x, x0) =δp(λ)(−1)|λ 0| ζλ+(x)ϕ+λ(p)(x 0) +ζ− λ(x)ϕ − λ(p)(x 0)−1 2ξλ(x)ψλ(p)(x 0) =1 2δp(λ)(−1) |λ0| ζλ+(x)−ζλ−(x) ϕ+λ(p)(x 0)−ϕ− λ(p)(x 0) , because ξλ(x) = ζλ+(x) +ζ − λ(x) and ψλ(p)(x0) = ϕ+λ(p)(x 0) +ϕ− λ(p)(x 0) by Clifford theory.
Denote by π = (π0, . . . , πp−1) the cyclic structure of x0. Assume that x is a
p-regular element andx0 ∈ D. Since/ xhas type λ, it follows that λ(p)
0 = ∅ and
`(λ(0p)) = 0, and π06=∅. Furthermore, any conjugate ofxalso has typeλ, thus we
ϕ+λ(p)(x
0)−ϕ−
λ(p)(x
0) = 0andB(x, x0) = 0. Assume nowxisp-singular andx0 ∈ D,
that isπ0 =∅. Then`(λ (p)
0 )>0. Again, the formula for character induction and
the proof of [24, Lemma 8.9] giveϕ+λ(p)(x
0)−ϕ−
λ(p)(x
0) = 0. Hence,B(x, x0) = 0.
Finally, we proved that ifF(x, x0)6= 0, thenxand x0 are either both p-regular elements and inD, or bothp-singular and not inD. SinceJbhas the same property by part (i) of the theorem, we deduce that this is also the case for IbA. By [9, Prop. 2.15], we deduce that IA is a perfect isometry with respect to the set of
p-regular elements ofAne andD.
The result now follows from the same process as in part (i). The analogue of the perfect isometryI2 of case (i) is the perfect isometry betweenB0∗ andB given
in [9, Thm 4.21].
Remark 4.8. If w < p, then Livesey shows that the isometry I0 of the proof
is a Broué isometry. Hence using [9, Thm 4.21], Lemma 4.3, Remark 4.4 and Theorem 4.5, shows that the isometry given in Theorem 4.7(i) is a Broué isometry. Furthermore, still in the case where w < p, the argument given by Livesey in the proof of [24, Thm. 10.2] carries forward to show that F, and thus IbA, satisfy Property (i) of a Broué isometry. SinceIbA also satisfies Property (ii), this similarly implies that the isometry given in Theorem 4.7(ii) is a Broué isometry.
Corollary 4.9. If B is any spinp-block of Sne or of Ane of p-weight w, then the
set of characters of B labeled by bar partitions λsuch thatλ0 =∅ is a p-basic set
of B.
Proof. In the case of Sn, this follows from Theorem 4.7, Theorems 3.3 and 3.4,e and [8, Prop 2.2] whenw >0. Ifw= 0, thenB contains a single character and the result is obvious. IfB is spinp-block ofAne , then [9, Thm 4.21] provides a perfect isometry from B to a spin p-block of same p-weight w of some Sem, and which preserves the labeling of characters. The result then follows from the first part of
the proof and [8, Prop 2.2].
Any spin p-block of Sne or of Ane has a p-basic set by Corollary 4.9, and any other block has a p-basic set by [8]. Hence, Sne and Ane have p-basic sets, which proves Theorem 1.1.
Remark 4.10. Recall thatSnhas another, non-isomorphic, double covering group, denoted bySn. In fact, we can derive from [26] thatb Snb ⊂CSn. More precisely,e there is a bijection ϕ :Sne → Snb , t 7→r(t)t where r(t) is some 4th-root of unity depending on the sign of the permutationθ(t). The bijectionϕyields the inclusion
b
Sn ⊂ CSn, which in turn implies that the two group algebras coincide, that ise A=CSnb =CSn. In particular, to any irreducible representatione ρ:A →End(V), we can associate irreducible representations ρ|
e
Sn and ρ|Sbn of Sne and Snb with
charactersχ
e
SnandχSbn, respectively. ConsiderΨ :CIrr(Sne )→CIrr(Snb )defined
byχ
e
Sn7→χSbn. Then for allt∈Sen, we have Ψ(χ
e
Sn)(ϕ(t)) =χSbn(ϕ(t)) = tr(ρ(ϕ(t)), V) =r(t) tr(ρ(t), V) =r(t)χSen(t).
By linearity, we obtain that, for allχ∈CIrr(Sne )and allt∈Sn,e
Let pbe an odd prime number. Let t ∈Sn. Sincee r(t)is a 2-element, t isp -regular if and only ifϕ(t)is, and we obtain from this last equality and the definition ofχ7→χbthat, for allχ∈Irr(Sne ),
[
Ψ(χ) = Ψ(χ).b
HenceΨis a perfect isometry with respect to thep-regular elements ofSenandSbn, and by [8, Prop 2.2],Sen has ap-basic set if and only ifSbn has one.
5. Some further remarks
First notice that, by construction, thep-basic setbS ofSen we produce is stable under tensor product byε. Furthermore, when restricted toAen, it gives ourp-basic setbAofAen, that is, the elements of bA are exactly the irreducible constituents of the restrictions toAen of the elements ofbS.
By Remark 3.5, and using the perfect isometry (constructed in [9, Thm 5.12]) between a p-block ofAn and Irr(H)with respect to p-regular elements of An and
θ(D), one can directly obtain the p-basic set of An constructed in [8, Thm 5.2]. The advantage of this new method is that we need not worry about Property (2) in [8, Thm 1.1] which is automatically satisfied.
IfB is any spinp-block of Sen or Aen, then the characters in thep-basic setbof
B obtained in Corollary 4.9 are labeled by bar partitionsλsuch thatλ(p)= (λ0=
∅, λ1, . . . , λ(p−1)/2), or, equivalently, by the(p−1)/2-quotients ofw. Furthermore, for each suchλ, one has
σ(λ(p)) = (−1)w−`(λ0)= (−1)w,
so that all quotients label two spin characters ofSen and one ofAne ifwis odd and
σ(B) = 1, or ifwis even andσ(B) =−1, and one spin character ofSne and two of e
An iswis odd andσ(B) =−1, or ifwis even andσ(B) = 1. In particular, all the characters inbare self-associate, or none is. Sincebisε-stable, the proof of [8, Prop 6.1] implies that the same is true for the set of irreducible Brauer characters inB, that is, all are self-associate or none is. As a consequence, we also obtain a formula for the number of irreducible Brauer characters inB. We therefore recover results of Olsson (see [30, Prop (3.21) and (3.22)]) obtained by sophisticated methods.
We end this paper with some considerations about what we like to callOlsson’s Principle. Roughly speaking, Olsson’s Principle predicts that, for any statement which holds for Sn, an analogous spin statement must hold forSen, even though one might have to be careful with signs along the proof.
We now present a few illustrations of this principle which appear in our work. Recall that the irreducible characters ofSnare parametrized by the set of partitions ofn. For any primep, a partitionλofn is uniquely determined by itsp-coreλ(p)
and its p-quotient λ(p) = (λ1, . . . , λp) which is a p-tuple of partitions whose sizes add up tow, the p-weight ofλ. The celebrated Nakayama Conjecture states that two irreducible charactersχλ andχµ ofSn belong to the samep-block if and only ifλ(p)=µ(p). The spin analogue is given by theMorris Conjecture(Theorem 4.6).
Now, the p-basic set of Sn constructed in [8] consists of those characters which are labeled by partitions λ with λ(p+1)/2 = ∅, i.e. without symmetric diagonal
(p)-hooks. The spin analogue of this is the set of bar partitions λ ofn such that
enlightened by thedoubling construction presented in [29]. Via this process, a bar partitionλofnis associated to a partitionD(λ)of2nsuch thatλ0=∅if and only ifD(λ)(p+1)/2=∅.
The way the irreducible spin characters ofGe can be labeled byp-quotients ofw, just like the irreducible characters of Gare labeled byp-quotients ofw is another example of Olsson’s Principle. It should therefore not come as a surprise that the perfect isometry of Theorem 4.7 (which associates spin characters of Sne and
e
G corresponding to the same p-quotient of w), is precisely the spin analogue of the perfect isometry of [8, Thm 3.6] (which associates characters of Sn and G
corresponding to the samep-quotient ofw), including the signδp(λ)(−1)|λ
0|
which is the spin analogue ofδp(λ)(−1)|λ
(p+1)/2|
.
By the same token, the basic set of SI(G)e we construct in Theorem 3.3 is the spin analogue of that ofGgiven in [8, Thm 4.3].
It should be noted that, while Olsson’s Principle can be an inspiration for results aboutSen, it is by no means a proof. However, it can be a useful guide to indicate what the proof should be. This is indeed the narrative behind the sequence given by [9], [24] and the present paper to generalize to the spin case the proof of [8], whose result’s spin analogue was inspired to the authors by Olsson’s Principle several years ago.
Acknowledgements. Part of this work was done at the CIRM in Luminy during a research in pairs stay. The authors wish to thank the CIRM gratefully for their financial and logistical support. The first author is supported by Agence Nationale de la Recherche Projet ACORT ANR-12-JS01-0003. The second author also acknowledges financial support from the Engineering and Physical Sciences Research Council grantCombinatorial Representation TheoryEP/M019292/1. The authors wish to thank the referee for several helpful suggestions.
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Université Paris-Diderot Paris 7, Institut de mathématiques de Jussieu – Paris Rive Gauche, UFR de mathématiques, Case 7012, 75205 Paris Cedex 13, France.
E-mail address:[email protected]
Institute of Mathematics, University of Aberdeen, King’s College, Fraser Noble Building, Aberdeen AB24 3UE, UK
