Irreducible tensor products of representations of covering groups of symmetric and alternating groups

在特征3下仍保持不可约的对称群与交错群的不可约自旋表示

Irreducible spin representations of symmetric and alternating groups which remain irreducible in characteristic 3

本文研究对称群与交错群真双重覆盖的普通不可约自旋表示在特征3下何时仍不可约或齐次，综合归纳限制、次数计算和射影特征分解等方法给出除一类自旋RoCK块相关划分外的分类。

The Isaacs-Navarro Conjecture for covering groups of the symmetric and alternating groups in odd characteristic

In this paper, we prove that a refinement of the Alperin-McKay Conjecture for $p$-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, whenever $p$ is an odd prime.

Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups

Young's orthogonal basis is a classical basis for an irreducible representation of a symmetric group. This basis happens to be a Gelfand-Tsetlin basis for the chain of symmetric groups. It is well-known that the chain of alternating groups, just like the chain of symmetric groups, has multiplicity-free restrictions for irreducible representations. Therefore each irreducible representation of an alternating group also admits Gelfand-Tsetlin bases. Moreover, each such representation is either the restriction of, or a subrepresentation of, the restriction of an irreducible representation of a sym…

Irreducible projective representations of the alternating group which remain irreducible in characteristic 2

For any finite group G it is an interesting question to ask which ordinary irreducible representations of G remain irreducible in a given characteristic p. We answer this question for p=2 when G is the proper double cover of the alternating group. As a key ingredient in the proof, we prove a formula for the decomposition numbers in Rouquier blocks of double covers of symmetric groups, in terms of Schur P-functions.

Representations of the alternating group which are irreducible over subgroups. II

We prove that non-trivial representations of the alternating group $A_n$ are reducible over a primitive proper subgroup which is isomorphic to some alternating group $A_m$.

交错群的循环特征标

Cyclic Characters of Alternating Groups

本文研究交错群元素在不可约表示中的特征值与循环表示分解，刻画含非零不变向量的配对，并构造新的全局共轭类族以重新证明 Heide–Zalessky 结论。

On the irreducible representation algebra of the alternating group of degree four

We obtain a description of the irreducible representation algebra of the alternating group of degree four over the ring of 2-adic integers.

The irreducible characters of the alternating Hecke algebras

This paper computes the irreducible characters of the alternating Hecke algebras, which are deformations of the group algebras of the alternating groups. More precisely, we compute the values of the irreducible characters of the semisimple alternating Hecke algebras on a set of elements indexed by minimal length conjugacy class representatives and we show that these character values determine the irreducible characters completely. As an application we determine a splitting field for the alternating Hecke algebras in the semisimple case.

Irreducible restrictions of representations of alternating groups in small characteristics: Reduction theorems

We study irreducible restrictions from modules over alternating groups to subgroups. We get reduction results which substantially restrict the classes of subgroups and modules for which this is possible. This is known when the characteristic of the ground field is greater than $3$, but the small characteristics cases require a substantially more delicate analysis and new ideas. In view of our earlier work on symmetric groups we may consider only the restriction of irreducible modules over alternating groups which do not extend to symmetric groups. This work fits into the Aschbacher-Scott progr…

Irreducible restrictions of representations of symmetric and alternating groups in small characteristics

Building on reduction theorems and dimension bounds for symmetric groups obtained in our earlier work, we classify the irreducible restrictions of representations of the symmetric and alternating groups to proper subgroups. Such classification is known when the characteristic of the ground field is greater than $3$, but the small characteristics cases require a substantially more delicate analysis and new ideas. Our results fit into the Aschbacher-Scott program on maximal subgroups of finite classical groups.

对称群与交错群不可约表示中置换的特征值研究

On eigenvalues of permutations in irreducible representations of symmetric and alternating groups

本文研究对称群与交错群不可约表示中置换的特征值问题，刻画了复数域上对称群不可约表示下置换的特征值集合，并将结果推广至交错群情形。

对称群与交错群的旋表示不可约限制

Irreducible restrictions of spin representations of symmetric and alternating groups

本文研究代数闭域上几乎拟单群中，限制不可约的子群与模的分类问题，并聚焦于交错群与对称群的 Schur 双覆盖情形。

Basic sets for the double covering groups of the symmetric and alternating groups in odd characteristic

In this paper, following the methods of http://arxiv.org/abs/1305.7449v2, we show that the double covering groups of the symmetric and alternating groups have p-basic sets for any odd prime p.

On the tensor square of irreducible representations of reductive Lie superalgebras

For semisimple Lie superalgebras over an algebraically closed field of characteristic zero, whose category of finite dimensional super representations is semisismple, we classify all irreducible super representations for which the alternating or symmetric square representation is irreducible or decomposes into an irreducible representation and a trivial representation.

Semisimple representations of alternating cyclotomic Hecke algebras

We define alternating cyclotomic Hecke algebras in higher levels as subalgebras of cyclotomic Hecke algebras under an analogue of Goldman's hash involution. We compute the rank of these algebras and construct a full set of irreducible representations in the semisimple case, generalising Mitsuhashi's results.

交错群在特征2中与线性特征成比例的旋量特征

Spin characters of the alternating group which are proportional to linear characters in characteristic 2

分类了交错群的旋量与非旋量不可约特征在2模约化下成比例的情况，等价于在奇阶元上成比例的情况。

Alternating quaternary algebra structures on irreducible representations of sl(2,C)

We determine the multiplicity of the irreducible representation V(n) of the simple Lie algebra sl(2,C) as a direct summand of its fourth exterior power $Λ^4 V(n)$. The multiplicity is 1 (resp. 2) if and only if n = 4, 6 (resp. n = 8, 10). For these n we determine the multilinear polynomial identities of degree $\le 7$ satisfied by the sl(2,C)-invariant alternating quaternary algebra structures obtained from the projections $Λ^4 V(n) \to V(n)$. We represent the polynomial identities as the nullspace of a large integer matrix and use computational linear algebra to find the canonical basis of th…

A note on restriction of Characters of Alternating groups to Sylow Subgroups

We restrict irreducible characters of alternating groups of degree divisible by $p$ to their Sylow $p$-subgroups and study the number of linear constituents.

对称群若干不可约特征的覆盖数

Covering Numbers of Some Irreducible Characters of the Symmetric Group

本文确定了对称群中由若干两行划分和钩形划分标记的不可约特征的覆盖数，涵盖奇偶 n 的不同情形及其共轭划分。

Labelling the character tables of symmetric and alternating groups

Let $X$ be a character table of the symmetric group $S_n$. It is shown that unless $n = 4$ or $n=6$, there is a unique way to assign partitions of $n$ to the rows and columns of $X$ so that for all $λ$ and $ν$, $X_{λν}$ is equal to $χ^λ(ν)$, the value of the irreducible character of $S_n$ labelled by $λ$ on elements of cycle type $ν$. Analogous results are proved for alternating groups, and for the Brauer character tables of symmetric and alternating groups.

Huppert's Conjecture for Alternating groups

We prove that the alternating groups of degree at least $5$ are uniquely determined up to an abelian direct factor by the degrees of their irreducible complex representations. This confirms Huppert's Conjecture for alternating groups.

On Quasi Steinberg characters of Symmetric and Alternating groups and their Double Covers

An irreducible character of a finite group $G$ is called quasi $p$-Steinberg character for a prime $p$ if it takes a nonzero value on every $p$-regular element of $G$. In this article, we classify the quasi $p$-Steinberg characters of Symmetric ($S_n$) and Alternating ($A_n$) groups and their double covers. In particular, an existence of a non-linear quasi $p$-Steinberg character of $S_n$ implies $n \leq 8$ and of $A_n$ implies $n \leq 9$.

Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

The partition algebra $\mathsf{P}_k(n)$ and the symmetric group $\mathsf{S}_n$ are in Schur-Weyl duality on the $k$-fold tensor power $\mathsf{M}_n^{\otimes k}$ of the permutation module $\mathsf{M}_n$ of $\mathsf{S}_n$, so there is a surjection $\mathsf{P}_k(n) \to \mathsf{Z}_k(n) := \mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k}),$ which is an isomorphism when $n \ge 2k$. We prove a dimension formula for the irreducible modules of the centralizer algebra $\mathsf{Z}_k(n)$ in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities …

Sign conjugacy classes of the alternating groups

A conjugacy class $C$ of a finite group $G$ is a sign conjugacy class if every irreducible character of $G$ takes value 0, 1 or -1 on $C$. In this paper we classify the sign conjugacy classes of alternating groups.

Regular orbits of symmetric and alternating groups

Given a finite group $G$ and a faithful irreducible $FG$-module $V$ where $F$ has prime order, does $G$ have a regular orbit on $V$? This problem is equivalent to determining which primitive permutation groups of affine type have a base of size 2. In this paper, we classify the pairs $(G,V)$ for which $G$ has a regular orbit on $V$ where $G$ is a covering group of a symmetric or alternating group and $V$ is a faithful irreducible $FG$-module such that the order of $F$ is prime and divides the order of $G$.

The q-analogue of the alternating groups and its representations

We construct a subalgebra of the Hecke algebra of type A. This is a generalization of the group algebra of the alternating groups. All the equivalent classes of irreducible representations of the subalgebra and the q-analogue of the branching rule between symmetric groups and alternating groups is obtained.

Branching Rules for $n$-fold Covering Groups of $\mathrm{SL}_2$ over a Non-Archimedean Local Field

Let $\widetilde{G}$ be the $n$-fold covering group of the special linear group of degree two, over a non-Archimedean local field. We determine the decomposition into irreducibles of the restriction of the principal series representations of $\widetilde{G}$ to a maximal compact subgroup of $\widetilde{G}$.

Generating Functions on Covering Groups

In this paper we prove Conjecture 1.2 in \cite{B-F}. This enables us to establish the meromorphic continuation of the standard partial $L$ function $L^S(s,π^{(n)})$. Here, $π^{(n)}$ is a genuine irreducible cuspidal representation of the group $GL_r^{(n)}({\bf A})$.

On unitriangular basic sets for symmetric and alternating groups

We study the modular representation theory of the symmetric and alternating groups. One of the most natural ways to label the irreducible representations of a given group or algebra in the modular case is to show the unitriangularity of the decomposition matrices, that is, the existence of a unitriangular basic set. We study several ways to obtain such sets in the general case of a symmetric algebra. We apply our results to the symmetric groups and to their Hecke algebras and thus obtain new ways to label the simple modules for these objects. Finally, we show that these sets do not always exis…

Small-dimensional projective representations of symmetric and alternating groups

We classify the irreducible projective representations of symmetric and alternating groups of minimal possible and second minimal possible dimensions, and get a lower bound for the third minimal dimension. On the way we obtain some new results on branching which might be of independent interest.

一类有限维代数表示簇的不可约分量

Irreducible components of varieties of representations of a class of finite dimensional algebras

针对一类三次截断且带额外关系的非交换多项式代数，本文通过证明固定根分层或底座分层子簇的不可约性并比较两类覆盖，确定了其表示簇的不可约分量。

含除一个特征值外其余特征值重数均为1的非标量元的交错型线性群

Linear groups of alternating type containing non-scalar elements with all but one eigenvalues of multiplicity 1

该研究确定了交错群、对称群及其泛中心扩张中，含除一个特征值外其余特征值重数均为1的非标量元的不可约表示，基域为任意特征的代数闭域。

Covering $\mathsf{Irrep}(S_n)$ With Tensor Products and Powers

We study when a tensor product of irreducible representations of the symmetric group $S_n$ contains all irreducibles as subrepresentations; we say such a tensor product covers $\mathsf{Irrep}(S_n)$. Our results show that this behavior is typical. We first give a general sufficient criterion for tensor products to have this property, which holds asymptotically almost surely for constant-sized collections of (Plancherel or uniformly) random irreducibles. We also consider the minimal tensor power of a single fixed irreducible representation needed to cover $\mathsf{Irrep}(S_n)$. Here a simple low…

McKay graphs for alternating and classical groups

Let $G$ be a finite group, and $α$ a nontrivial character of $G$. The McKay graph $\mathcal{M}(G,α)$ has the irreducible characters of $G$ as vertices, with an edge from $χ_1$ to $χ_2$ if $χ_2$ is a constituent of $αχ_1$. We study the diameters of McKay graphs for finite simple groups $G$. For alternating groups, we prove a conjecture made in [LST]: there is an absolute constant $C$ such that $\hbox{diam}\,{\mathcal M}(G,α) \le C\frac{\log |\mathsf{A}_n|}{\log α(1)}$ for all nontrivial irreducible characters $α$ of $\mathsf{A}_n$. Also for classsical groups of symplectic or orthogonal type of …

Specht modules decompose as alternating sums of restrictions of Schur modules

Schur modules give the irreducible polynomial representations of the general linear group $\mathrm{GL}_t$. Viewing the symmetric group $\mathfrak{S}_t$ as a subgroup of $\mathrm{GL}_t$, we may restrict Schur modules to $\mathfrak{S}_t$ and decompose the result into a direct sum of Specht modules, the irreducible representations of $\mathfrak{S}_t$. We give an equivariant Möbius inversion formula that we use to invert this expansion in the representation ring for $\mathfrak{S}_t$ for $t$ large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternat…

The Imprimitive Faithful Complex Characters of the Schur Covers of the Symmetric and Alternating Groups

Using combinatorics and character theory, we determine the imprimitive faithful complex characters, i.e., the irreducible faithful complex characters which are induced from proper subgroups, of the Schur covers of the symmetric and alternating groups. Furthermore, for every imprimitive character we establish all its minimal block stabilizers. As a corollary, we also determine the monomial faithful characters of the Schur covers.

Characters of the irreducible representations with fundamental highest weight for the symplectic group in characteristic p

Let K be an algebraically closed field of characteristic p>0 and let Sp(2m) be the symplectic group of rank m over K. The main theorem of this article gives the character of the rational simple Sp(2m)-modules with fundamental highest weight as an explicit alternating sum of characters of Weyl modules. One obtains several formulae for the dimensions of these simple modules, what allows us to investigate the asymptotic behavior of these dimensions, for a given p, when the rank m is growing towards infinity. One also gets the simple Weyl modules with fundamental highest weight, and the article en…

Unitriangular basic sets for blocks of the symmetric and alternating groups of small weights

We study the existence of unitriangular basic sets for the symmetric group which behave nicely with respect to the Mullineux involution. Such sets give a natural labelling for the modular irreducible representations. We show that, for any odd prime p, the p-blocks of the symmetric group with weight 2 have stable unitriangular basic sets which we describe by studying the combinatorics of partitions in these blocks.

对称群和交错群双覆盖中每个元素的最小多项式

Minimal Polynomials in Spin Representations of Symmetric and Alternating Groups

本文研究了对称群和交错群双覆盖中元素的最小多项式，探讨其在不可约旋转变换表示中的性质。

Alternating sums over pi-subgroups

Dade's conjecture predicts that if p is a prime, then the number of irreducible characters of a finite group of a given p-defect is determined by local subgroups. In this paper we replace $p$ by a set of primes pi and prove a pi-version of Dade's conjecture for pi-separable groups. This extends the (known) p-solvable case of the original conjecture and relates to a pi-version of Alperin's weight conjecture previously established by the authors.

关于斜群代数的不可约态射

On the Irreducible Morphisms for Skew group algebras

本文引入 Galois 半覆盖函子研究斜群代数的模范畴，完整刻画其不可约态射与几乎可裂序列，并证明稳定秩在斜化下保持，进而确定斜温和代数的稳定秩。

A theorem of Mœglin-Waldspurger for covering groups

Let $E$ be a non-Archimedian local field of characteristic zero and residue characteristic $p$. Let ${\bf G}$ be a connected reductive group defined over $E$ and $π$ an irreducible admissible representation of $G={\bf G}(E)$. A result of C. Mœglin and J.-L. Waldspurger (for $p \neq 2$) and S. Varma (for $p=2$) states that the leading coefficient in the character expansion of $π$ at the identity element of ${\bf G}(E)$ gives the dimension of a certain space of degenerate Whittaker forms. In this paper we generalize this result of Mœglin-Waldspurger to the setting of covering groups $\tilde{G}$ …

Classification of irreducible representations of metaplectic covers of the general linear group over a non-archimedean local field

Let $F$ be a non-archimedean local field. The classification of the irreducible representations of $GL_n(F)$, $n\ge0$ in terms of supercuspidal representations is one of the highlights of the Bernstein--Zelevinsky theory. We give an analogous classification for metaplectic coverings of $GL_n(F)$, $n\ge0$.

Branching laws for the metaplectic cover of ${\rm GL}_{2}$

Let $F$ be a non-Archimedian local field of characteristic zero and $E/F$ a quadratic extension. The aim of the present article is to study the multiplicity of an irreducible admissible representation of ${\rm GL}_2(F)$ occurring in an irreducible admissible genuine representation of non-trivial two fold covering $\widetilde{\rm GL}_2(E)$ of ${\rm GL}_2(E)$.

关于对称群和交错群的二面体分支系数的正性

On the Positivity of Dihedral Branching Coefficients of the Symmetric and Alternating Groups

研究确定了对称群和交错群的二面体分支系数的正性条件，并推广了关于循环子群的已知结果。

对称群与交错群在特征3下成比例的旋量特征

Spin characters of symmetric and alternating groups which are proportional in characteristic 3

研究有限群G的p-模约化中，两个不可约表示何时成比例的问题，特别针对p=3时双覆盖群的旋量特征，给出了完整分类。

Composition factors of 2-parts spin representations of symmetric groups

Given an odd prime $p$, we identify composition factors of the reduction modulo $p$ of spin irreducible representations of the covering groups of symmetric groups indexed by partitions with 2 parts and find some decomposition numbers.

Irreducible characters of finite simple groups constant at the p-singular elements

In representation theory of finite groups an important role is played by irreducible characters of p-defect 0, for a prime p dividing the group order. These are exactly those vanishing at the p-singular elements. In this paper we generalize this notion investigating the irreducible characters that are constant at the p-singular elements. We determine all such characters of non-zero defect for alternating, symmetric and sporadic simple groups. We also classify the irreducible characters of quasi-simple groups of Lie type that are constant on the non-identity unipotent elements. In particular, w…

$\mathrm{PSL}_2(q)$ 的特征标覆盖数

Character covering number of $\mathrm{PSL}_2 (q)$

本文研究了群 $\mathrm{PSL}_2(q)$ 的特征标覆盖数，证明了当 $q\geq 8$ 时，其所有非平凡不可约特征标 $χ$ 的三次或四次幂均可覆盖全部不可约特征标集合。

由 $l'$-类上的值确定的交错群和对称群的常特征标和射影特征标

Characters and projective characters of alternating and symmetric groups determined by values on $l'$-classes

本文确定了交错群及其双重覆盖群中在 $l'$-类上取值相同的普通不可约特征标对，指出这些特征标对由特定分区标记，并探讨了 $l$ 为素数及 $l=3$ 时的情形。

对称群与交错群特征标的次数及素数幂阶零点

Degrees and prime power order zeros of characters of symmetric and alternating groups

本文研究对称群与交错群不可约特征标的相关性质，证明对称群特征标次数p-部分由p-幂阶消失元决定，交错群p=2时仅存在两种可能，并推广至p-块亏度与特征标p-高度。

The Weyl Character Formula, the half-spin representations, and equal rank subgroups

Let B be a reductive Lie subalgebra of a semi-simple Lie algebra of the same rank both over the complex numbers. To each finite dimensional irreducible representation $V_λ$ of F we assign a multiplet of irreducible representations of B with m elements in each multiplet, where m is the index of the Weyl group of B in the Weyl group of F. We obtain a generalization of the Weyl character formula; our formula gives the character of $V_λ$ as a quotient whose numerator is an alternating sum of the characters in the multiplet associated to $V_λ$ and whose denominator is an alternating sum of the char…

对称群特征标的覆盖数

Covering numbers for characters of symmetric groups

本文研究对称群不可约特征标的幂的覆盖性质，证明当 n>4 时，任意非线性不可约特征标的 k 次幂覆盖全部不可约特征标当且仅当 k≥n-1。

Unitary Representations and Heisenberg Parabolic Subgroup

In this paper, we study the restriction of an irreducible unitary representation $π$ of the universal covering $\widetilde{Sp}_{2n}(\mb R)$ to a Heisenberg maximal parabolic group $\tilde P$. We prove that if $π|_{\tilde P}$ is irreducible, then $π$ must be a highest weight module or a lowest weight module. This is in sharp constrast with the $GL_n(\mathbb R)$ case. In addition, we show that for a unitary highest or lowest weight module, $π|_{\tilde P}$ decomposes discretely. We also treat the groups $U(p,q)$ and $O^*(2n)$.