Representations Of Symmetric Group Research Articles

Multiplicity of the trivial representation in rank-selected homology of the partition lattice

We study the multiplicity bS(n) of the trivial representation in the symmetric group representations βS on the (top) homology of the rank-selected partition lattice ΠnS. We break the possible rank sets S into three cases: (1) 1∉S, (2) S=1,…,i for i⩾1, and (3) S=1,…,i,j1,…,jl for i,l⩾1, j1>i+1. It was previously shown by Hanlon that bS(n)=0 for S=1,…,i. We use a partitioning for Δ(Πn)/Sn due to Hersh to confirm a conjecture of Sundaram [S. Sundaram, The homology representations of the symmetric group on Cohen–Macaulay subposets of the partition lattice, Adv. Math. 104 (1994) 225–296] that bS(n)>0 for 1∉S. On the other hand, we use the spectral sequence of a filtered complex to show bS(n)=0 for S=1,…,i,j1,…,jl unless a certain type of chain of support S exists. The partitioning for Δ(Πn)/Sn allows us then to show that a large class of rank sets S=1,…,i,j1,…,jl for which such a chain exists do satisfy bS(n)>0. We also generalize the partitioning for Δ(Πn)/Sn to Δ(Πn)/Sλ; when λ=(n−1,1), this partitioning leads to a proof of a conjecture of Sundaram about (S1×Sn−1)-representations on the homology of the partition lattice.

The finite basis property for some classes of irreducible representations of symmetric groups

The possibility of defining irreducible representations of symmetric groups by means of finitely many relations is studied. The existence of finite bases is established for the classes of representations corresponding to two-part partitions and to partitions in the fundamental alcove.

Lexicographic Shellability for Balanced Complexes

We introduce a notion of lexicographic shellability for pure, balanced boolean cell complexes, modelled after the CL-shellability criterion of Bjorner and Wachs (Adv. in Math. 43 (1982), 87–100) for posets and its generalization by Kozlov (Ann. of Comp. 1(1) (1997), 67–90) called CC-shellability. We give a lexicographic shelling for the quotient of the order complex of a Boolean algebra of rank 2n by the action of the wreath product S2 w Sn of symmetric groups, and we provide a partitioning for the quotient complex Δ(Πn)/Sn. Stanley asked for a description of the symmetric group representation βS on the homology of the rank-selected partition lattice ΠnS in Stanley (J. Combin. Theory Ser. A 32(2) (1982), 132–161), and in particular he asked when the multiplicity bS(n) of the trivial representation in βS is 0. One consequence of the partitioning for Δ(Πn)/Sn is a (fairly complicated) combinatorial interpretation for bS(n)s another is a simple proof of Hanlon's result (European J. Combin. 4(2) (1983), 137–141) that b1,…,i(n) e 0. Using a result of Garsia and Stanton from (Adv. in Math. 51(2) (1984), 107–201), we deduce from our shelling for Δ(B2n)/S2 w Sn that the ring of invariants k[x1,…,x2n]S2 w Sn is Cohen-Macaulay over any field k.

Projective representations of symmetric groups via Sergeev duality

In this article, we determine the irreducible projective representations of the symmetric group Sd and the alternating group Ad over an algebraically closed field of characteristic p 6= 2. These matters are well understood in the case p = 0, thanks to the fundamental work of Schur [24] in 1911, as well as the much more recent work of Nazarov [19, 20], Sergeev [25, 26] and others. So the focus here is primarily on the case of positive characteristic, where surprisingly little is known as yet. In particular, we obtain a natural combinatorial labelling of the irreducibles in terms of a certain set RPp(d) of restricted p-strict partitions of d. Such partitions arose recently in work of Kashiwara, Miwa, Peterson and Yung [11] and Leclerc and Thibon [14] on the q-deformed Fock space of the affine Kac-Moody algebra of type A p−1. Leclerc and Thibon proposed that RPp(d) should label the irreducible projective representations in some natural way, and we show here how this can be done. Note that for p = 3, 5, the labelling problem was solved in [1, 3], while if p = 2 all projective representations of Sd and Ad are linear so do not need to be considered further here. To be more precise, recall that λ is a partition of d if λ = (λ1, λ2, . . . ) is a non-increasing sequence of non-negative integers summing to d. Call λ p-strict if in addition

Relating spin representations of symmetric and hyperoctahedral groups

The aim of this paper is to present recent results of Yamaguchi (A duality of twisted group algebras of symmetric groups and Lie superalgebras, Ph.D. Thesis, University of Tokyo, 1998), on the structure of a certain twisted group algebra of the hyperoctahedral group considered by Sergeev (Math. Sb. 51 (1985) 419–425 (English translation)) in terms of a twisted group algebra of the symmetric group corresponding to a nontrivial cocycle. This result allows one to explain a mysterious dependence of both descriptions of the corresponding characters in the works of Sergeev (Math. Sb. 51 (1985) 419–425 (English translation)) and Schur (J. Reine Angew. Math. 139 (1911) 155–250) on the so-called Schur Q-functions.

On Tensor Products of Modular Representations of Symmetric Groups

On Tensor Products of Modular Representations of Symmetric Groups

Kawanaka Invariants for Representations of Weyl Groups

Let W be a Weyl group and let V be the natural CW-module, i.e., the reflection representation. For a complex irreducible character χ of W, we consider the invariantIχ;q≔|W|−1∑w∈Wχw2det1+qw|V/det1−qw|Vintroduced by N. Kawanaka. We determine I(χ;q) explicitly. Looking over these results, we observe a relation between Kawanaka's invariants I(χ;q) and the two-sided cells. For example, if a two-sided cell consists of a single element χ, then the Kawanaka invariant I(χ;q) can be expressed as ∏li=1(1+qhi)/(1−qhi) with some integers hi. This expression can be regarded as a quantization of the usual hook formula for the dimension of irreducible representations of symmetric groups.

The Howe duality and the projective representations of symmetric groups

The symmetric group S k \mathfrak {S}_{k} possesses a nontrivial central extension, whose irreducible representations, different from the irreducible representations of S k \mathfrak {S}_{k} itself, coincide with the irreducible representations of the algebra A k \mathfrak {A}_{k} generated by indeterminates τ i , j \tau _{i, j} for i ≠ j i\neq j , 1 ≤ i , j ≤ n 1\leq i, j\leq n subject to the relations τ i , j = − τ j , i , τ i , j 2 = 1 , τ i , j τ m , l = − τ m , l τ i , j if { i , j } ∩ { m , l } = ∅ ; τ i , j τ j , m τ i , j = τ j , m τ i , j τ j , m = − τ i , m for any i , j , l , m . \begin{gather*} \tau _{i, j}=-\tau _{j, i}, \quad \tau _{i, j}^{2}=1, \quad \tau _{i, j}\tau _{m, l}=-\tau _{m, l}\tau _{i, j}\text { if }\{i, j\}\cap \{m, l\}=\emptyset ;\ \tau _{i, j}\tau _{j, m}\tau _{i, j}=\tau _{j, m}\tau _{i, j}\tau _{j, m}=-\tau _{i, m}\; \; \text { for any } i, j, l, m. \end{gather*} Recently M. Nazarov realized irreducible representations of A k \mathfrak {A}_{k} and Young symmetrizers by means of the Howe duality between the Lie superalgebra q ( n ) \mathfrak {q}(n) and the Hecke algebra H k = S k ∘ C l k H_{k}=\mathfrak {S}_{k}\circ Cl_{k} , the semidirect product of S k \mathfrak {S}_{k} with the Clifford algebra C l k Cl_{k} on k k indeterminates. Here I construct one more analog of Young symmetrizers in H k H_{k} as well as the analogs of Specht modules for A k \mathfrak {A}_{k} and H k H_{k} .

Asymptotic Results on Modular Representations of Symmetric Groups and Almost Simple Modular Group Algebras

In fact, this is a question about infinite simple groups because it is easy for G finite, and because a non-trivial normal subgroup gives rise to a non-trivial ideal different from the augmentation ideal. Also note that the problem reduces easily to the question on when the augmentation ideal is simple as a ring. The first interesting class of groups with almost simple FG was discovered in [3]. This class is rather exotic and contains groups like the universal Hall group and algebraically closed groups. For locally finite groups G a substantial progress was achieved recently by using representation theory of finite groups, see [18, 19, 16], and others. The representation theory approach transforms the problems on ideals to certain problems on asymptotic behavior of representations of finite groups, which are often of independent interest. The method is more effective over fields of characteristic zero as the theory of ordinary representations is much better elaborated than the modular theory. This paper is devoted to the modular aspect of the theory. One of our main results deals with the case where G is a direct limit of alternating groups and F is any field of characteristic p > 2. We note that the case where charF = 0 was considered in [18]. Let N be the set of natural numbers. Denote by Alt(Ω) and Sym(Ω) (or, simply, An and Σn) the alternating and symmetric groups, respectively, on a set Ω with |Ω| = n. Let Alt(Ω1) ⊂ Alt(Ω2) ⊂ · · · ⊂ Alt(Ωi) ⊂ . . . (1)

The globally irreducible representations of symmetric groups

Let $K$ be an algebraic number field and $\mathcal {O}$ be the ring of integers of $K$. Let $G$ be a finite group and $M$ be a finitely generated torsion free $\mathcal {O} G$-module. We say that $M$ is a globally irreducible $\mathcal {O} G$-module if, for every maximal ideal $\mathfrak {p}$ of $\mathcal {O}$, the $k_\mathfrak {p} G$-module $M\otimes _{ \mathcal {O}} k_\mathfrak {p}$ is irreducible, where $k_\mathfrak {p}$ stands for the residue field $\mathcal {O}/\mathfrak {p}$. Answering a question of Pham Huu Tiep, we prove that the symmetric group $\Sigma _n$ does not have non-trivial globally irreducible modules. More precisely we establish that if $M$ is a globally irreducible $\mathcal {O} \Sigma _n$-module, then $M$ is an $\mathcal {O}$-module of rank $1$ with the trivial or sign action of $\Sigma _n$.

Maximal ideals in modular group algebras of the finitary symmetric and alternating groups

The main result of the paper is a description of the maximal ideals in the modular group algebras of the finitary symmetric and alternating groups (provided the characteristic p of the ground field is greater than 2). For the symmetric group there are exactly p 1 such ideals and for the alternating group there are (p 1)/2 of them. The description is obtained in terms of the annihilators of certain systems of the 'completely splittable' irreducible modular representations of the finite symmetric and alternating groups. The main tools used in the proofs are the modular branching rules (obtained earlier by the second author) and the 'Mullineux conjecture' proved recently by FordKleshchev and Bessenrodt-Olsson. The results obtained are relevant to the theory of PI-algebras. They are used in a later paper by the authors and A. E. Zalesskii on almost simple group algebras and asymptotic properties of modular representations of symmetric groups.

Representations of Symmetric Groups and Free Probability

We consider representations of symmetric groupsSqfor largeq. We give the asymptotic behaviour of the characters when the corresponding Young diagrams, rescaled by a factorq−1/2, converge to some prescribed shape. This behaviour can be expressed in terms of the free cumulants for a probability measure associated with the limit shape of the diagram. We also show that the basic operations of representation theory, like taking tensor products, restriction, or induction, have a limiting behavior which can be described using the free probability theory of D. Voiculescu.

Nilpotent Orbit Varieties and the Atomic Decomposition of theQ-Kostka Polynomials

AbstractWe study the coordinate rings ofscheme-theoretic intersections of nilpotent orbit closures with the diagonal matrices. Here μ′ gives the Jordan block structure of the nilpotent matrix. de Concini and Procesi [5] proved a conjecture of Kraft [12] that these rings are isomorphic to the cohomology rings of the varieties constructed by Springer [22, 23]. The famousq-Kostka polynomialis the Hilbert series for the multiplicity of the irreducible symmetric group representation indexed by λ in the ring. Lascoux and Schützenberger [15, 13] gave combinatorially a decomposition ofas a sum of “atomic” polynomials with non-negative integer coefficients, and Lascoux proposed a corresponding decomposition in the cohomology model.Our work provides a geometric interpretation of the atomic decomposition. The Frobenius-splitting results of Mehta and van der Kallen [19] imply a direct-sum decomposition of the ideals of nilpotent orbit closures, arising from the inclusions of the corresponding sets. We carry out the restriction to the diagonal using a recent theorem of Broer [3]. This gives a direct-sum decomposition of the ideals yielding the, and a new proof of the atomic decomposition of theq-Kostka polynomials.

Branching Rules for Modular Representations of Symmetric Groups, IV

Branching Rules for Modular Representations of Symmetric Groups, IV

Q-Deformed Fock spaces and modular representations of spin symmetric groups

We use the Fock space representation of the quantum affine algebra of type $A^{(2)}_{2n}$ to obtain a description of the global crystal basis of its basic level 1 module. We formulate a conjecture relating this basis to decomposition matrices of spin symmetric groups in characteristic $2n+1$.

Gap conditions for representations of symmetric groups

For the symmetric groups S( n) with n ≥ 6 we construct representations which satisfy a strong gap condition for the dimensions of various fixed point sets, as desired in equivariant surgery, and a condition on the isotropy groups, as used in the construction of manifolds with interesting fixed point sets. This allows us to construct smooth actions of S( n) on sphere such that the fixed point set is exactly one point, or any given closed, stably parallelizable manifold, all of whose components have the same dimension.

Applications de Gauss et pléthysme

The irreducible representations of Gl (n,ℂ) can be described by Schur functors, the composition of which defines plethysm. Its understanding is an important problem of invariant theory, as well as in relation with the representations of symmetric groups.

Hecke algebras at roots of unity and crystal bases of quantum affine algebras

We present a fast algorithm for computing the global crystal basis of the basic $$U_q (\widehat{\mathfrak{s}\mathfrak{l}}_n )$$ -module. This algorithm is based on combinatorial techniques which have been developed for dealing with modular representations of symmetric groups, and more generally with representations of Hecke algebras of typeA at roots of unity. We conjecture that, upon specializationq→1, our algorithm computes the decomposition matrices of all Hecke algebras at an th root of 1.

Branching Rules for Modular Representations of Symmetric Groups III: Some Corollaries and a Problem of Mullineux

Branching Rules for Modular Representations of Symmetric Groups III: Some Corollaries and a Problem of Mullineux

Quantum affine algebras and affine Hecke algebras

One of the most beautiful results from the classical period of the representation theory of Lie groups is the correspondence, due to Frobenius and Schur, between the representations of symmetric groups and those of general or special linear groups. If V0 is the natural irreducible (n+ 1)–dimensional representation of SLn+1(CI), the symmetric group Sl acts on V ⊗l 0 by permuting the factors. This action obviously commutes with the action of SLn+1(CI). It follows that one may associate to any right Sl–module M a representation of SLn+1(CI), namely FS(M) = M⊗SlV ⊗l 0 , the action of SLn+1(CI) on FS(M) being induced by its natural action on V ⊗l 0 . The main result of the Frobenius–Schur theory is that, if l ≤ n, the assignment M → FS(M) defines an equivalence from the category of finite–dimensional representations of Sl to the category of finite–dimensional representations of SLn+1(CI), all of whose irreducible components occur in V ⊗l 0 . Around 1985, Drinfeld and Jimbo independently introduced a family of Hopf algebras Uq(g), depending on a parameter q ∈ CI , associated to any symmetrizable Kac–Moody algebra g. Assuming that q is not a root of unity, Jimbo [7] proved an analogue of the Frobenius–Schur correspondence in which SLn+1(CI) is replaced by Uq(sln+1), V0 by the natural (n+1)–dimensional irreducible representation V of Uq(sln+1), and Sl by its Hecke algebra Hl(q 2). In [5], Drinfeld announced an analogue of the Frobenius–Schur theory for the Yangian Y (sln+1), which is a “deformation” of the universal enveloping algebra of the Lie algebra of polynomial maps CI → sln+1. The role of Sl in this theory is played by the degenerate affine Hecke algebra Λl, an algebra whose defining relations are obtained from those of the affine Hecke algebra Ĥl(q 2) by letting q → 1 in a certain non–trivial fashion. 1 Both authors were partially supported by the NSF, DMS-9207701.