Hook Partitions Research Articles

On the decomposition of the Foulkes module

The Foulkes module \({H^{(a^b)}}\) is the permutation module for the symmetric group S ab given by the action of S ab on the collection of set partitions of a set of size ab into b sets each of size a. The main result of this paper is a sufficient condition for a simple \({\mathbb{C} S_{ab}}\) -module to have zero multiplicity in \({H^{(a^b)}}\) . A special case of this result implies that no Specht module labelled by a hook partition (ab − r, 1r) with r ≥ 1 appears in \({H^{(a^b)}}\) .

Tensors, matchings and codes

We exploit the structure of the critical orbital sets of symmetry classes of tensors associated to sign uniform partitions and we establish new connections between symmetry classes of tensors, matchings on bipartite graphs and coding theory. In particular, we prove that the orthogonal dimension of the critical orbital sets associated to single hook partitions λ=(w,1n-w) equals the value of the coding theoretic function A(n,4,w). When w=2 we reobtain this number as the independence number of the Dynkin diagram An-1.

Path tableaux and combinatorial interpretations of immanants for class functions on $S_n$

Let $χ ^λ$ be the irreducible $S_n$-character corresponding to the partition $λ$ of $n$, equivalently, the preimage of the Schur function $s_λ$ under the Frobenius characteristic map. Let $\phi ^λ$ be the function $S_n →ℂ$ which is the preimage of the monomial symmetric function $m_λ$ under the Frobenius characteristic map. The irreducible character immanant $Imm_λ {(x)} = ∑_w ∈S_n χ ^λ (w) x_1,w_1 ⋯x_n,w_n$ evaluates nonnegatively on each totally nonnegative matrix $A$. We provide a combinatorial interpretation for the value $Imm_λ (A)$ in the case that $λ$ is a hook partition. The monomial immanant $Imm_{{\phi} ^λ} (x) = ∑_w ∈S_n φ ^λ (w) x_1,w_1 ⋯x_n,w_n$ is conjectured to evaluate nonnegatively on each totally nonnegative matrix $A$. We confirm this conjecture in the case that $λ$ is a two-column partition by providing a combinatorial interpretation for the value $Imm_{{\phi} ^λ} (A)$. Soit $χ ^λ$ le caractère irréductible de $S_n$ qui correspond à la partition λ de l'entier n, ou de manière équivalente, la préimage de la fonction de Schur $s_λ$ par l'application caractéristique de Frobenius. Soit $\phi ^λ$ la fonction $S_n →ℂ$ qui est la préimage de la fonction symétrique monomiale m_λ . La valeur du caractère irréductible immanent $Imm_λ {(x)} = ∑_w ∈S_n χ ^λ (w) x_1,w_1 ⋯x_n,w_n$ est non négative pour chaque matrice totalement non négative. Nous donnons une interprétation combinatoire de la valeur $Imm_λ (A)$ lorsque $λ$ est une partition en équerre. Stembridge a conjecturé que la valeur de l'immanent monomial $Imm_{{\phi} ^λ} (x) = ∑_w ∈S_n φ ^λ (w) x_1,w_1 ⋯x_n,w_n$ de $\phi ^λ$ est elle aussi non négative pour chaque matrice totalement non négative. Nous confirmons cette conjecture quand λ satisfait $λ _1 ≤2$, et nous donnons une interprétation combinatoire de $Imm_{{\phi} ^λ} (A)$ dans ce cas.

On the support varieties of tilting modules

Let G be a reductive algebraic group scheme defined over the finite field F p , with Frobenius kernel G 1 . The tilting modules of G are defined as rational G -modules for which both the module itself and its dual have good filtrations. In 1997, J.E. Humphreys conjectured that the support varieties of certain tilting modules for regular weights should be given by the Lusztig bijection between cells of the affine Weyl group and nilpotent orbits of G , when p > h , where h is the Coxeter number. We present a conjecture for the support varieties of tilting modules when G = G L n . Our conjecture is equivalent to Humphreys’ conjecture for p ≥ h and regular weights, but our formulation allows us to consider small p or singular weights as well. We obtain results for several infinite classes of tilting modules, including the case p = 2 , and tilting modules whose support variety corresponds to a hook partition. In the case p = 2 , we prove the conjecture by S. Donkin for the support varieties of tilting modules.

The complexity of the Specht modules corresponding to hook partitions

We show that the complexity of the Specht module corresponding to any hook partition is the p-weight of the partition. We calculate the variety and the complexity of the signed permutation modules. Let E s be a representative of the conjugacy class containing an elementary abelian p-subgroup of a symmetric group generated by s disjoint p-cycles. We give formulae for the generic Jordan types of signed permutation modules restricted to E s and of Specht modules corresponding to hook partitions μ restricted to E s where s is the p-weight of μ.

Resolutions of De Concini–Procesi Ideals of Hooks

We find a minimal generating set for the defining ideal of the schematic intersection of the set of diagonal matrices with the closure of the conjugacy class of a nilpotent matrix indexed by a hook partition. The structure of this ideal allows us to compute its minimal free resolution and give an explicit description of the graded Betti numbers, and study its Hilbert series and regularity.

Garsia–Haiman modules for hook partitions and Green polynomials with two variables

We consider Garsia–Haiman modules for the symmetric group, a doubly graded generalization of Springer modules. Our main interest lies in singly graded submodules of a Garsia–Haiman module. We show that these submodules satisfy a certain combinatorial property, and verify that this property is implied by a behavior of Macdonald polynomials at roots of unity.

Green vertices and sources of simple modules of the symmetric group labelled by hook partitions

We determine the Green vertices and sources of many of the simple modular representations of the finite symmetric group being parametrized by hook partitions.

Enumeration of Generalized Hook Partitions

Enumeration of Generalized Hook Partitions

Elementary divisors of Specht modules

Let ℋ q ( S n ) be the Iwahori-Hecke algebra of the symmetric group defined over the ring Z [ q , q − 1 ] . The q -Specht modules of ℋ q ( S n ) come equipped with a natural bilinear form. In this paper we try to compute the elementary divisors of the Gram matrix of this form (which need not exist since Z [ q , q − 1 ] is not a principal ideal domain). When they are defined, we give the relationship between the elementary divisors of the Specht modules S q ( λ ) and S q ( λ ′ ) , where λ ′ is the conjugate partition. We also compute the elementary divisors when λ is a hook partition and give examples to show that in general elementary divisors do not exist.

A v-analogue of Peel's theorem

We compute the v-decomposition numbers d λ μ ( v ) for λ being a hook partition, and μ e-regular.

Embeddings of higher Lie modules

The higher Lie modules of the general linear group GL( V) over a finite dimensional vector space V arise naturally from the Poincaré–Birkhoff–Witt basis of the tensor algebra over V. They are indexed by partitions. For the higher Lie modules corresponding to hook partitions of n a complete chain of embeddings is obtained. As an application, a new inductive proof of Klyachko's result on the irreducible components of the classical Lie module is given. Additionally, all irreducible components of multiplicity 1 are determined.

A generating set of Solomon's descent algebra

Abstract Solomon's descent algebra is generated by sums of descent classes corresponding to certain hook shapes. This particularly implies that the ring of class functions of any finite symmetric group S n is generated by the irreducible characters corresponding to certain hook partitions of n . As another consequence, a second generating set of Solomon's descent algebra (and of the ring of class functions of S n ) is obtained related to the major index of permutations.

Immant dominance orderings for hook partitions

Immant dominance orderings for hook partitions

On homomorphisms between weyl modules for hook partitions

Let K be an infinite field of finite characteristic p. We denote by Sk (n,r) the Schur Algebra [4. §4] for n and r. The aim of this paper is to determine Hom Sk(n,r)( Va V a-1) where Va is the Weyl module associated with the hook partition