The Isaacs–Navarro conjecture for covering groups of the symmetric and alternating groups in odd characteristic

DOI 10.1007/s10801-011-0277-5

The Isaacs–Navarro conjecture for covering groups of the symmetric and alternating groups in odd characteristic

Jean-Baptiste Gramain

Received: 7 July 2010 / Accepted: 20 January 2011 / Published online: 8 February 2011

© Springer Science+Business Media, LLC 2011

Abstract In this paper, we prove that a refinement of the Alperin–McKay Conjecture forp-blocks of finite groups, formulated by I.M. Isaacs and G. Navarro in 2002, holds for all covering groups of the symmetric and alternating groups, wheneverpis an odd prime.

Keywords Representation theory·Symmetric group·Covering groups· Bar-partitions

1 Introduction

In order to understand properties of thep-modular representation theory of a finite groupG, one often tries to reduce to a problem about thep-local subgroups ofG, i.e., the normalizers of itsp-subgroups. This is illustrated by many results, such as Brauer’s three Main Theorems, and several conjectures, such as Broué’s Abelian De- fect Conjecture, Dade’s Conjecture or the Alperin–McKay Conjecture.

I.M. Isaacs and G. Navarro have formulated in [5] some refinements of the McKay and Alperin–McKay Conjectures for arbitrary finite groups. Consider a fi- nite groupGand a prime p. LetB be a p-block ofG, with defect group D, and letb be the Brauer correspondent ofB inN_G(D). Throughout this paper, we will use a p-valuation ν on Z, given by ν(n)=a if n=p^aq with (p, q)=1. The heighth(χ )∈Z≥0 of an irreducible (complex) characterχ∈B is then defined by the equalityν(χ (1))=ν(|G|)−ν(|D|)+h(χ ). We denote byM(B)andM(b)the sets of characters of height 0 ofB andb, respectively. The Alperin–McKay Con- jecture then asserts that|M(B)| = |M(b)|(while the McKay Conjecture states that

J.-B. Gramain (

Institut de Mathématiques de Jussieu, UFR de Mathématiques, Université Denis Diderot, Paris VII, 2 place Jussieu, 75251 Paris Cedex 05, France

e-mail:gramain@math.jussieu.fr

|M(G)| = |M(N_G(P ))|, whereP ∈Syl_p(G), andM(G)andM(N_G(P ))denote the sets of irreducible characters ofp-degree ofGandNG(P ),respectively).

In [5], Isaacs and Navarro predicted that something stronger must happen, namely that this equality can be refined when considering thep-parts of the character de- grees. For anyn∈N, we writen=npn_p, withnp=p^ν(n). For any 1≤k≤p−1, we define subsets M_k(B) andM_k(b) of M(B) andM(b),respectively, by letting Mk(B)= {χ∈M(B);χ (1)_p ≡ ±k (modp)}andMk(b)= {ϕ∈M(b);ϕ(1)_p ≡

±k (modp)}. We then have the following

Conjecture 1.1 [5, Conjecture B] For 1≤k≤p−1, we have|Mck(B)| = |Mk(b)|, wherec= [G:N_G(D)]p.

Note that Conjecture1.1obviously implies the Alperin–McKay Conjecture (by letting k run through {1, . . . , p−1}), but also implies another refinement of the McKay Conjecture; if we letM_k(G)= {χ∈Irr(G);χ (1)≡ ±k (modp)}then, by considering all blocks ofGwith defect groupP ∈Syl_p(G), we obtain|M_k(G)| =

|M_k(N_G(P ))|, since[G:N_G(P )] ≡1(modp)(see [5, Conjecture B]).

Isaacs and Navarro proved Conjecture 1.1 whenever D is cyclic, or G is p- solvable or sporadic. P. Fong proved it for symmetric groupsS(n)in [2], and R. Nath for alternating groupsA(n)in [8]. In this paper, we prove that Conjecture1.1holds in all the covering groups of the symmetric and alternating groups, provided p is odd (Theorem5.1). The proof makes heavy use of the powerful combinatorics under- lying the representation theory of these groups. In particular, Conjecture1.1comes from an explicit bijection, given in terms of the bar-partitions used to parametrize the irreducible characters.

In Sect.2, we present the covering groupsS⁺(n)andS⁻(n)and their irreducible characters, first studied by I. Schur in [11], as well as theirp-blocks. It turns out that the main work to be done is on so-called spin blocks. We also give various results on the degrees of spin characters, generalizing the methods used by Fong in [2]. Most of these results are of a combinatorial nature, and the concepts they involve are also pre- sented here. Section3is devoted to proving Theorem3.4which reduces the problem to proving only that Conjecture1.1holds for the principal spin block ofS⁺(pw).

This reduction theorem is a refinement of [6, Theorem 2.2] that G.O. Michler and J.B. Olsson proved in order to establish that the Alperin–McKay Conjecture holds for covering groups. Finally, the case of the principal spin block ofS⁺(pw)is treated in Sect.4.

2 Covering groups

In this section, we introduce the objects and preliminary results we will need about covering groups and their characters. Unless stated otherwise, the following results can be found in [6].

2.1 Covering groups

For any integern≥1, I. Schur has defined (by generators and relations) two central extensionsS(n)ˆ andS(n)˜ of the symmetric group S(n)(see [11], p. 164). We have

S(1)ˆ ∼= ˜S(1)∼=Z/2Z, and, forn≥2, there is a nonsplit exact sequence 1−→ z −→ ˆS(n)−→^π S(n)−→1,

where z =Z(S(n))ˆ ∼=Z/2Z.

Whenevern≥4, these two extensions are non-isomorphic, except whenn=6.

However, they are isoclinic, so that their representation theory is virtually the same.

Hence, for our purpose, it is sufficient to study one of them. Throughout this paper, we will writeS⁺(n)forS(n).ˆ

IfH is a subgroup ofS(n), we letH⁺=π⁻¹(H )andH⁻=π⁻¹(H ∩A(n)).

In particular,H⁻has index 1 or 2 inH⁺, andH⁺=H⁻if and only ifH⊂A(n).

We defineS⁻(n)=A(n)⁻=A(n)⁺. HenceS⁻(n)is a central extension ofA(n)of degree 2.

The groupsA(6)andA(7)also have one 6-fold cover each, which, together with the above groups, give all the covering groups ofS(n)andA(n).

2.2 Characters, blocks and twisted central product

From now on, we fix an odd primep. For anyH ≤S(n), the irreducible complex characters ofH^εfall into two categories: those that havezin their kernel, and which can be identified with those of H (if ε=1) or those of H ∩A(n) (if ε= −1), and those that don’t have z in their kernel. These (faithful) characters are called spin characters. We denote by SI(H^ε)the set of spin characters ofH^ε, and we let SI0(H^ε)=SI(H^ε)∩M(H^ε)(with the notation of Sect.1).

If B is a p-block of H^ε then, because p is odd, it is known that either B ∩ SI(H^ε)= ∅orB⊂SI(H^ε), in which case we say thatBis a spin block ofH^ε.

Any twoχ , ψ∈Irr(H^ε)are called associate ifχ↑^H+=ψ↑^H+(ifε= −1) or if χ↓H⁻=ψ↓H⁻ (ifε=1). Then each irreducible character ofH^ε has exactly 1 or 2 associate characters. Ifχ is itself its only associate, we say thatχ is self-associate (written s.a.), we putχ^a=χand letσ (χ )=1. Otherwise,χ has a unique associate ψ=χ; we say thatχ is non-self-associate (written n.s.a.), we put χ^a=ψand we letσ (χ )= −1.

If H⁺ =H⁻, then χ ∈Irr(H⁺) and ϕ ∈Irr(H⁻) are said to correspond if χ , ϕ↑^H+H⁺=0. In this case, Clifford’s theory implies thatσ (χ )= −σ (ϕ).

IfH1, H2, . . . , Hk≤S(n)act (non-trivially) on disjoint subsets of{1, . . . , n}, then one can define the twisted central productH⁺=H₁⁺ˆ×··· ˆ×H_k⁺≤S⁺(n)(see [11]

or [3]). Then|H⁺| =₂k¹−1|H₁⁺| · |H₂⁺| · · · |H_k⁺| =2|H1| · |H2| · · · |Hk|. Also, one ob- tains SI(H⁺)from the SI(H_i⁺)’s as follows:

Proposition 2.1 (See [11, §28]) There is a surjective map ˆ⊗:

SI(H₁⁺)× · · · ×SI(H_k⁺)−→SI(H⁺), (χ1, . . . , χk)−→χ1ˆ⊗··· ˆ⊗χk,

which satisfies the following properties. Suppose χ_i, ψ_i ∈SI(H_i⁺)for 1≤i≤k.

Then

(i) σ (χ₁ˆ⊗··· ˆ⊗χ_k)=σ (χ₁)· · ·σ (χ_k), and(χ₁ˆ⊗··· ˆ⊗χ_k)(1)=2^s/2χ₁(1)· · · × χ_k(1), wheresis the number of n.s.a. characters in{χ₁, . . . , χ_k}and denotes integral part.

(ii) χ1ˆ⊗··· ˆ⊗χk andψ1ˆ⊗··· ˆ⊗ψk are associate if and only ifχi andψi are asso- ciate for alli.

(iii) χ₁ˆ⊗··· ˆ⊗χ_k=ψ₁ˆ⊗··· ˆ⊗ψ_kif and only ifχ_i andψ_i are associate for alliand [σ (χ1)· · ·σ (χ_k)=1] or [σ (χ1)· · ·σ (χ_k)= −1 and|{i|χ_i=ψ_i}|is even].

2.3 Partitions and bar-partitions

Just as the irreducible characters ofS(n)are parametrized by the partitions ofn, the spin characters ofS⁺(n)have a combinatorial description. We letP (n) be the set of all partitions ofn, andP₀(n)be the subset of all partitions in distinct parts, also called bar-partitions. We writeλnforλ∈P (n), andλnforλ∈P₀(n). We also write, in both cases,|λ| =n.

It is well known that Irr(S(n))= {χ_λ, λn}. For anyλn, we write h(λ)for the product of all hook-lengths inλ. We then haveh(λ)=h_λ,ph_λ,p, where h_λ,p (respectively, h_λ,p) is the product of all hook-lengths divisible byp (prime top, respectively) inλ. The Hook-Length Formula then givesχ_λ(1)=_h(λ)^n! .

If we remove all the hooks of length divisible bypinλ, we obtain itsp-coreλ_(p). The information onp-hooks is stored in thep-quotientλ^(p)ofλ. Ifn=pw+r, with λ(p)r, thenλ^(p)is ap-tuple of partitions ofw, i.e.,λ^(p)=(λ⁽⁰⁾, . . . , λ^(p−1))and

|λ⁽⁰⁾| + · · · + |λ^(p−1)| =w. The partitionλis uniquely determined by itsp-core and p-quotient. Also, for any integerk, there exists a (canonical) bijection between the kp-hooks inλand thek-hooks inλ^(p)(i.e., in theλ⁽ⁱ⁾’s).

Finally, the Nakayama Conjecture states that χ_λ, χ_μ∈Irr(S(n)) belong to the samep-block if and only ifλandμhave the samep-core.

We now present the analogue properties for bar-partitions and spin characters. For any bar-partitionλ=(a₁, . . . , a_m)ofn, witha₁>· · ·> a_m>0, we letm(λ)=m, and define the sign ofλbyσ (λ)=(−1)^n−m(λ). We then have

Theorem 2.2 (See [11, §41]) For each signε∈ {1,−1}, there is a (canonical) sur- jective mapf^ε:SI(S^ε(n))−→P0(n)such that:

(i) σ (χ )=εσ (f^ε(χ ))for allχ∈SI(S^ε(n)).

(ii) For anyχ , ψ∈SI(S^ε(n)), we havef^ε(χ )=f^ε(ψ )if and only ifχandψare associate.

(iii) Ifχ∈SI(S⁺(n))andϕ∈SI(S⁻(n)), thenf⁺(χ )=f⁻(ϕ)if and only ifχand ϕcorrespond.

In particular, eachλn labels one s.a. characterχ or two associate characters χ andχ^a. Throughout this paper, we will denote by λ the set of spin characters labeled by λ, and write (abusively) λ ∈SI(S^ε(n)), and λ(1)for the (common) degree of any spin character in λ. We will also sometimes write λ+to emphasize that λ ∈SI(S⁺(n))(and λ₋if λ ∈SI(S⁻(n))).

For the following results on bars, cores and quotients, we refer to [9]. For any odd integerq, lete=(q−1)/2. We define aq¯-quotient of weightwto be any tuple of

partitions(λ⁽⁰⁾, λ⁽¹⁾, . . . , λ^(e))such that λ⁽⁰⁾∈P₀(w₀), λ⁽ⁱ⁾∈P (w_i)for 1≤i≤e, and w0+w1+ · · · +w_e=w. We define its sign by σ ((λ⁽⁰⁾, λ⁽¹⁾, . . . , λ^(e)))= (−1)^w−w0σ (λ⁽⁰⁾).

Now take any bar-partitionλ=(a1, . . . , a_m)ofnas above. The bars inλcan be read in the shifted Young diagramS(λ)ofλ. This is obtained from the usual Young diagram ofλby shifting theith rowi−1 positions to the right. Thejth node in the ith row is called the(i, j )-node, and corresponds to the barB_ij. The bar-lengths in theith row are obtained by writing (from left to right inS(λ)) the elements of the following set in decreasing order:{1,2, . . . , ai} ∪ {a_i+a_j|j > i} \ {a_i−a_j|j > i}.

The bars are of three types:

• Type 1. These are barsB_ijwithi+j ≥m+2 (i.e., in the right part ofS(λ)). They are ordinary hooks inS(λ), and their lengths are the elements of{1,2, . . . , ai− 1} \ {a_i−a_j|j > i}.

• Type 2. These are barsB_ij withi+j =m+1 (in particular, the corresponding nodes all belong to the same column ofS(λ)). Their length is preciselyai, and the bar is all of theith row ofS(λ).

• Type 3. The lengths{ai+aj|j > i}correspond to barsBij withi+j ≤m. The bar consists of theith row together with thejth row ofS(λ).

Bars of type 1 and 2 are called unmixed, while those of type 3 are called mixed. The unmixed bars inλcorrespond exactly to the hooks in the partitionλ^∗, which admits as aβ-set the set of parts ofλ.

For anyλn, we writeh(λ)¯ for the product of all bar-lengths inλ. We then have h(λ)¯ = ¯h_λ,ph¯_λ,p, whereh¯_λ,p (respectively, h¯_λ,p) is the product of all bar-lengths divisible byp(prime top,respectively) inλ. We then have the following analogue of the Hook-Length Formula (proved by A.O. Morris [7, Theorem 1])

λ(1)=2^{(n−m(λ))/2} n! h(λ)¯ .

If we remove all the bars of length divisible byp inλ, we obtain itsp-core¯ λ_(p)¯ (which is still a bar-partition), and itsp-quotient¯ λ^(p)¯. Ifn=pw+r, withλ_(p)¯ r, thenλ^(p)¯ is ap-quotient of weight¯ w in the sense defined above. The bar-partition λis uniquely determined by itsp-core and¯ p-quotient. Also, for any integer¯ k, there exists a canonical bijection between the set ofkp-bars inλand the set ofk-bars in λ^(p)¯ (where ak-bar inλ^(p)¯ =(λ⁽⁰⁾, λ⁽¹⁾, . . . , λ^((p−1)/2))is ak-bar inλ⁽⁰⁾or ak-hook in one ofλ⁽¹⁾, . . . , λ^((p−1)/2)).

The distribution of the spin characters ofS⁺(n)into spin blocks was first con- jectured for p odd by Morris. It was first proved by J.F. Humphreys in [4], then differently by M. Cabanes, who also determined the structure of the defect groups of spin blocks (see [1]).

Proposition 2.3 Letχ , ψ∈SI(S^ε(n))andpbe an odd prime. Thenχis ofp-defect 0 if and only iff^ε(χ )is ap-core. If¯ f^ε(χ )is not ap-core, then¯ χ andψbelong to the samep-block if and only iff^ε(χ )_(p)¯ =f^ε(ψ )_(p)¯ .

One can therefore define thep-core of a spin block¯ Band its weightw(B), as well as its signδ(B)=σ (f^ε(χ )_(p)¯)(for anyχ∈B). We then have

Proposition 2.4 (See [1]) IfB is a spin block ofS^ε(n)of weight w, then a defect groupXofBis a Sylowp-subgroup ofS^ε(pw).

2.4 Removal ofp-bars

The following result is the bar-analogue of [2, Lemma 3.2]; it describes how the removal ofp-bars affects the product ofp-bar-lengths.

Proposition 2.5 Supposeλnhasp-core¯ λ_(p)¯ . Then

h¯_λ,p≡ ±2^−a(λ)h¯_λ(p)¯,p= ±2^−a(λ)h(λ¯ _(p)¯ ) (modp), wherea(λ)is the number ofp-bars of type 3 to remove fromλto getλ(p)¯ .

Proof LetBij be ap-bar inλandλ−Bij be the bar-partition obtained fromλby removingB_ij. We distinguish two cases, depending on whether B_ij is unmixed or mixed.

First suppose thatBij is unmixed (i.e.,i+j > m(λ)). We start by examining the unmixedp-bars in λandλ−B_ij. These correspond, in the notation above, to the p-hooks inλ^∗ and(λ−B_ij)^∗,respectively (consideringλandλ−B_ij asβ-sets).

The set of parts ofλisX= {a1, . . . , am}, and the set of non-zero parts ofλ−Bij

isY = {a₁, . . . , a_i−1, a_i −p, a_i+1, . . . , a_m}(or Y = {a₁, . . . , a_i−1, a_i+1, . . . , a_m} if a_i =p). Thep-hooks inλ^∗ (resp.,(λ−B_ij)^∗) therefore correspond to pairs(x, y) with 0≤x < y,(y−x, p)=1, andx /∈X,y∈X(resp.,x /∈Y,y∈Y).

IfBij is of type 1 (i.e.,i+j > m(λ)+1), thenai−p >0, so that|Y| = |X| and(λ−B_ij)^∗=λ^∗−h for some p-hook h in λ. In this case, we are thus ex- actly in the same context as [2, Lemma 3.2], and we get h_λ∗,p ≡ ±h_λ∗−h,p =

±h_(λ−Bij)∗,p (modp). Note that the result of [2, Lemma 3.2] is, in fact, incorrect, as the right hand side should be multiplied by(−1)^μ/κ, whereμ/κis the relative sign associated toμandκ. The mistake is to be found in the proof, where the leg-length Lh of the hook removed should appear (four lines before the end), yielding, in our case,h_λ∗,p≡(−1)^Lh+1h_λ∗−h,p (modp).

If, on the other hand,B_ij is of type 2 (i.e.,i+j =m(λ)+1), thena_i−p=0, andY=X\ {p}. Note that, in this case,Y is not aβ-set for a partition of|λ^∗| −p, whileY∪ {0}is. Thep-hooks in(λ−Bij)^∗correspond to either pairs(x, y)with y=a_i, which also correspond top-hooks inλ^∗, or to pairs(p, y), withy > pand y∈X. These new hooks have lengths(a₁−p), . . . , (a_i−1−p). Finally, some hooks have disappeared: those corresponding to pairs(x, p)withx < pandx /∈X. These have lengths(p−x), for 0≤x < pandx /∈ {ai+1, . . . , am}.

We now turn to the mixedp-bars inλandλ−B_ij. Suppose first thatB_ij is of type 1. Thenm(λ)=m(λ−B_ij). Suppose that

a₁>· · ·> a_i−1> a_i+1>· · ·> a_k> a_i−p > a_k+1>· · ·> a_m.

To prove the result, we can simply ignore the bar-lengths which are common toλand λ−B_ij. The mixed bars which disappear when going fromλtoλ−B_ijhave lengths

(a₁+a_i), (a₂+a_i), . . . , (a_i−1+a_i) and

(a_i+a_i+1), (a_i+a_i+2), . . . , (a_i+a_m).

The mixed bars which appear have lengths

(a₁+a_i−p), (a₂+a_i−p), . . . , (a_i−1+a_i−p),

(ai+1+ai−p), . . . , (ak+ai−p) and (ai−p+ak+1), . . . , (ai−p+am).

If we then just consider the lengths not divisible byp, it is easy to see that we can pair the bars disappearing with those appearing. The pairs are of the form(b, b), where bis a bar inλandbis a bar inλ−B_ij, and|b| = |b| −p. We thus get, in this case,

bmixedp-bar inλ

|b| ≡

bmixedp-bar inλ−Bij

|b|(modp).

Together with the equality obtained above for unmixedp-bars, we obtain that, ifB_ij is ap-bar of type 1 inλ, thenh¯_λ,p≡ ± ¯h_λ−Bij,p(modp).

Now suppose thatB_ij is of type 2, i.e.,a_i=p. Then the mixed bars which disap- pear when going fromλtoλ−B_ij have lengths(a₁+p), (a₂+p), . . . , (a_i−1+p) (call theseA) and(p+a_i+1), (p+a_i+2), . . . , (p+a_m)(call theseB), while no new mixed bar appears.

The bars disappearing in A are compensated for by the hooks appearing in (λ−B_ij)^∗ in the study of unmixed bars above (since a_u+p≡a_u−p (mod p) for all 1≤u≤i−1, thep-parts are congruent modpwhen these are not divisible byp).

On the other hand, since 0< a_m<· · ·< a_i+1< a_i=p, all the bar-lengths inB are coprime top, and their product is

(p+a_i+1)(p+a_i+2)· · ·(p+a_m)≡a_i+1a_i+2· · ·a_m(modp).

Now the hooks disappearing in the above discussion of unmixed bars all have length prime topexcept one (corresponding tox=0). The product of the lengths prime to pis thus

0<x<p,x /∈{a_i+1,...,a_m}

(p−x)≡(−1)^p−1−m+i

0<x<p,x /∈{a_i+1,...,a_m}

x (modp).

Hence the product of thep-hook-lengths disappearing and thep-bar-lengths inBis congruent (modp) to

(−1)^p−1−m+i

0<y<p

y=(−1)^p−1−m+i(p−1)! ≡(−1)^p−m+i(modp)

(by Wilson’s Theorem). Finally, we obtain that, ifB_ij is ap-bar of type 2 inλ, then h¯_λ,p≡(−1)^p−m+ih¯_λ−Bij,p (modp).

We now suppose thatB_ij is ap-bar of type 3 inλ, i.e.,i < j,a_i > a_j anda_i+ a_j =p. The set of parts ofλisX= {a₁, . . . , a_m}and the set of parts ofλ−B_ij is Y= {a₁, . . . , a_i−1, a_i+1, . . . , a_j−1, a_j+1, . . . , a_m}. Ignoring as before the bars which

are common toλandλ−B_ij, we see that the unmixed bars which disappear fromλ toλ−B_ij have lengths

(ai−x)

0≤x < ai, x /∈ {am, . . . , ai+1} and (aj−x)

0≤x < aj, x /∈ {am, . . . , aj+1} , while those appearing have lengths

(a1−ai), . . . , (ai−1−ai), (a1−aj), . . . , (ai−1−aj) and (ai+1−aj), . . . , (aj−1−aj).

On the other hand, there is no mixed bar appearing, while the mixed bars disappearing have lengths

(a1+ai), . . . , (ai−1+ai) (rows 1,. . .,i−1, columni), (a_i+a_i+1), . . . , (a_i+a_j−1), (a_i+a_j), . . . , (a_i+a_m) (rowi),

(a1+a_j), . . . , (a_i−1+a_j) (rows 1, . . . , i−1, columnj), (a_i+1+a_j), . . . , (a_j−1+a_j) (rowsi+1,. . .,j−1, columnj), and

(a_j+a_j+1), . . . , (a_j+a_m) (rowj).

Now, sinceai+aj=p, we have, for any 1≤k≤m,

a_k−a_i≡a_k+a_j(modp) and a_k+a_i≡a_k−a_j(modp).

In particular,ak−ai (resp., ak+ai) is coprime top if and only ifak+aj (resp., ak−aj) is coprime top, and, in that case,

(a_k±a_i)_p=a_k±a_i≡a_k∓a_j≡(a_k∓a_j)_p(modp).

We thus have the following compensations between the appearing unmixed bars and the appearing mixed bars:

(a₁−a_i), . . . , (a_i−1−a_i) ←→ (a₁+a_j), . . . , (a_i−1+a_j) (a₁−a_j), . . . , (a_i−1−a_j) ←→ (a₁+a_i), . . . , (a_i−1+a_i) and (a_i+1−a_j), . . . , (a_j−1−a_j) ←→ (a_i+a_i+1), . . . , (a_i+a_j−1).

This accounts for all the appearing (unmixed) bars, and we’re left exactly with the following disappearing bar-lengths:

(a_i−x)

0≤x < a_i, x /∈ {a_m, . . . , a_i+1}

unmixed of type 1, (a_j−x)

0≤x < a_j, x /∈ {a_m, . . . , a_j+1}

unmixed of type 2,

(a_i+1+a_j), . . . , (a_j−1+a_j), (a_j+a_j+1), . . . , (a_j+a_m) mixed of type 1, (a_i+a_j+1), . . . , (a_i+a_m) mixed of type 2,

and(a_i+a_j)=pwhich can thus be ignored.

Now, for anyi+1≤k≤m,a_j+a_k=p−a_i+a_k≡ −(a_i−a_k) (modp), and, forj +1≤k≤m,a_i +a_k≡ −(a_j −a_k) (modp). Hence, taking the product, we obtain (modulop):

0≤x<a_i,x=a_j

(a_i−x)= ai! ai−aj

(type 1) and

0≤x<a_j

(a_j−x)=a_j!(type 2).

Nowa_j! =1·2· · ·a_j=(−1)^aj(−1)· · ·(−a_j)≡(−1)^aj(p−1)· · ·(p−a_j) (modp), so thata_j! ≡(−1)^aj(p−1)· · ·(a_i+1)ai (modp). We thus have, disappearing,

±aiai!(ai+1)· · ·(p−1)

a_i−a_j ≡ ± ai

a_i−a_j(p−1)! ≡ ∓ ai

a_i−a_j (modp) (this last equality being true by Wilson’s Theorem).

Finally,a_i−a_j=a_i−(p−a_i)≡ −2ai(modp), yielding a total of±2⁻¹(modp) disappearing (since, p being odd, 2 is invertible(modp), and ai < p so that we can simplify bya_i). We thus get that, ifB_ij is ap-bar of type 3 inλ, thenh¯_λ,p≡

±¹₂h¯_λ−Bij,p(modp).

Iterating the above results on all thep-bars to remove fromλto get to itsp-core¯ λ_(p)¯ , we finally obtain the desired equality, writinga(λ)for the number ofp-bars of type 3 to remove:

h¯_λ,p≡ ±2^−a(λ)h¯_λ(p)¯,p= ±2^−a(λ)h(λ¯ _(p)¯) (modp)

(since all the bars inλ_(p)¯ have length coprime top).

2.5 p-Core tower,¯ p-quotient tower and characters of¯ p-degree

In this section, we want to obtain an expression for the (value modulopof the)p- part of the degree of a spin character. We start by describing thep-core tower of a¯ bar-partition, introduced by Olsson in [9].

Take any λn. the p-core tower of¯ λ has rows R₀^λ, R₁^λ, R₂^λ, . . ., where the ith row R^λ_i contains one p-core and¯ (pⁱ −1)/2 p-cores (in particular, one can considerR^λ_i as a p¯ⁱ-quotient). We haveR^λ₀ = {λ(p)¯ }(the p-core of¯ λ). If thep-¯ quotient of λ is λ^(p)¯ =(λ⁽⁰⁾, λ⁽¹⁾, . . . , λ^(e)) (where e=(p−1)/2), then R^λ₁ = {λ⁽⁰⁾_(p)¯ , λ⁽¹⁾_(p), . . . , λ^(e)_(p)}. Writingλ^(0)(p)¯ =(λ^(0,0), λ^(0,1), . . . , λ^(0,e))the p-quotient of¯ λ⁽⁰⁾andλ^(i)(p)=(λ^(i,1), λ^(i,2), . . . , λ^(i,p))thep-quotient ofλ⁽ⁱ⁾(1≤i≤e), and tak- ing cores, we let

R₂^λ=

λ^(0,0)_(p)¯ , λ^(0,1)_(p) , . . . , λ^(0,e)_(p) , λ^(1,1)_(p) , . . . , λ^(1,p)_(p) , λ^(2,1)_(p) , . . . , λ^(e,p)_(p) .

Continuing in this way, we obtain thep-core tower of¯ λ. We define thep-quotient¯ tower of λ in a similar fashion: it has rows Q^λ₀, Q^λ₁, Q^λ₂, . . ., where the ith row

Q^λ_i contains onep-quotient and¯ (pⁱ −1)/2 p-quotients (in particular,Q^λ_i can be seen as a p¯ⁱ⁺¹-quotient). With the above notation, we have Q^λ₀= {λ^(p)¯ }, Q^λ₁= {λ^(0)(p)¯, λ^(1)(p), . . . , λ^(e)(p)}and

Q^λ₂=

λ^(0,0)(p)¯, λ^(0,1)(p), . . . , λ^(0,e)(p), λ^(1,1)(p), . . . , λ^(1,p)(p), λ^(2,1)(p), . . . , λ^(e,p)(p) . The following result will be useful later.

Lemma 2.6 Ifλnhasp-core tower¯ (R₀^λ, R₁^λ, . . . , R_m^λ), thenσ (λ)=_m

i=0σ (R^λ_i).

Proof We haveσ (λ)=σ (λ_(p)¯ )σ (λ^(p)¯ ), andσ (λ_(p)¯)=σ (R^λ₀).

Also,σ (λ^(p)¯ )=σ (λ⁽⁰⁾)(−1) ^i≥1|λ(i)|, and σ

λ⁽⁰⁾

=σ λ⁽⁰⁾_(p)¯

σ λ^(0)(p)¯

=σ Q^λ₀

=σ λ⁽⁰⁾_(p)¯

σ λ^(0,0)

(−1) ^{j≥1|λ(0,j )|}. Nowσ (R^λ₁)=σ (λ^(0)(p)¯)(−1) ^{i≥1|λ(i)(p)¯|}andσ (Q^λ₁)=σ (λ^(0,0))(−1) ^{i≥0,j≥1|λ(i,j )|}, so that

σ R₁^λ

σ Q^λ₁

=σ λ^(0)(p)¯

σ λ^(0,0)

(−1) ^{i≥1|λ(i)(p)¯|+ i≥0,j≥1|λ(i,j )|}

=σ λ⁽⁰⁾

(−1) ^{i≥1|λ(i)(p)¯|+ i,j≥1|λ(i,j )|}.

However, for eachi≥1, we have|λ⁽ⁱ⁾_(p)¯ | + _j≥1|λ^{(i,j )}| ≡ |λ⁽ⁱ⁾_(p)¯ | +p _j≥1|λ^{(i,j )}| (mod 2)(sincepis odd), and|λ⁽ⁱ⁾_(p)¯ | +p _j≥1|λ^{(i,j )}| = |λ⁽ⁱ⁾|. We therefore get

σ R^λ₁

σ Q^λ₁

σ λ⁽⁰⁾

(−1) ^i≥1|λ(i)|=σ λ^(p)¯

=σ Q^λ₀

.

Finally, we haveσ (λ)=σ (R₀^λ)σ (Q^λ₀), andσ (Q^λ₀)=σ (R^λ₁)σ (Q^λ₁), whenceσ (λ)= σ (R^λ₀)σ (R^λ₁)σ (Q^λ₁). Iterating this process, we deduce the result.

Now, writingβ_i(λ) for the sum of the cardinalities of the partitions inR^λ_i, one shows easily that|λ| = i≥0βi(λ)pⁱ (see [9]). Also, one gets the following bar- analogue of [2, Proposition 1.1]:

Proposition 2.7 [9, Proposition 3.1] In the above notation, νp

h(λ)¯

=n− i≥0βi(λ) p−1 .

In particular, λhasp-degree if and only if _i≥0β_i(λ)pⁱ is thep-adic decompo- sition ofn.

Letn= ^ki=0tipⁱbe thep-adic decomposition ofn. For each 0≤i≤k, letei= (pⁱ−1)/2, and writeR^λ_i = {μ⁽⁰⁾_i , μ⁽¹⁾_i , . . . , μ^(e_i ⁱ⁾}andQ^λ_i = {λ⁽⁰⁾_i , λ⁽¹⁾_i , . . . , λ^(e_i ⁱ⁺¹⁾}. Note thatQ^λ_k = {∅, . . . ,∅}.