Specht Modules Research Articles

Vertices for Iwahori–Hecke algebras and the Dipper–Du conjecture

Let $\mathscr{H}_n$ denote the Iwahori-Hecke algebra corresponding to the symmetric group $\mathfrak{S}_n$. We set up a Green correspondence for bimodules of these Hecke algebras, and a Brauer correspondence between their blocks. We examine Specht modules for $\mathscr{H}_n$ and compute the vertices of certain Specht modules, before using this to give a classification of the vertices of blocks of $\mathscr{H}_n$ in any characteristic. Finally, we apply this classification to resolve the Dipper-Du conjecture about the structure of vertices of indecomposable $\mathscr{H}_n$-modules.

First degree cohomology of Specht modules and extensions of symmetric powers

Let Σd denote the symmetric group of degree d and let K be a field of positive characteristic p. For p>2 we give an explicit description of the first cohomology group H1(Σd,Sp(λ)), of the Specht module Sp(λ) over K, labelled by a partition λ of d.We also give a sufficient condition for the cohomology to be non-zero for p=2 and we find a lower bound for the dimension.The cohomology of Specht modules has been considered in many papers including [10], [12], [15] and [21].Our method is to proceed by comparison with the cohomology for the general linear group G=GLn(K) and then to reduce to the calculation of ExtB1(SdE,Kλ), where B is a Borel subgroup of G, where SdE denotes the dth symmetric power of the natural module E for G and Kλ denotes the one dimensional B-module with weight λ.The main new input is the description of module extensions by: extensions sequences, coherent triples of extension sequences and coherent multi-sequences of extension sequences, and the detailed calculation of the possibilities for such sequences. These sequences arise from the action of divided powers elements in the negative part of the hyperalgebra of G.Our methods are valid also in the quantised context and we aim to treat this in a separate paper.

ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS

We investigate integral forms of certain simple modules over group algebras in characteristic 0 whose $p$-modular reductions have precisely three composition factors. As a consequence we, in particular, complete the description of the integral forms of the simple $\QQ\mathfrak{S}_n$-module labelled by the hook partition $(n-2,1^2)$. Moreover, we investigate the integral forms of the Steinberg module of finite special linear groups $\PSL_2(q)$ over suitable fields of characteristic 0. In the second part of the paper we explicitly determine the Solomon zeta functions of various families of modules and lattices over group algebra, including Specht modules of symmetric groups labelled by hook partitions and the Steinberg module of $\PSL_2(q)$.

Specht polynomials and modules over the Weyl algebra

In this paper we study the decomposition structure of a direct image of a polynomial ring under certain map.

Two-block Springer fibers of types C and D: a diagrammatic approach to Springer theory

We explain an elementary topological construction of the Springer representation on the homology of (topological) Springer fibers of types C and D in the case of nilpotent endomorphisms with two Jordan blocks. The Weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which appear in the definition of arc algebras of types B and D. We determine the decomposition of the representations into irreducibles and relate our construction to classical Springer theory. As an application we obtain presentations of the cohomology rings of all two-block Springer fibers of types C and D. Moreover, we deduce explicit isomorphisms between the Kazhdan-Lusztig cell modules attached to the induced trivial module and the irreducible Specht modules in types C and D.

Cohomology Classes of Interval Positroid Varieties and a Conjecture of Liu

To each finite subset of $\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture.However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes. To do this, we exhibit the appropriate diagram variety as a component in a degeneration of one of Knutson's interval positroid varieties (up to Grassmann duality). A priori, the cohomology classes of these interval positroid varieties are represented by affine Stanley symmetric functions. We give a different formula for these classes as ordinary Stanley symmetric functions, one with the advantage of being Schur-positive and compatible with inclusions between Grassmannians.

Specht modules labelled by hook bipartitions I

Brundan, Kleshchev and Wang equip the Specht modules Sλ over the cyclotomic Khovanov–Lauda–Rouquier algebra HnΛ with a homogeneous Z-graded basis. In this paper, we begin the study of graded Specht modules labelled by hook bipartitions ((n−m),(1m)) in level 2 of HnΛ, which are precisely the Hecke algebras of type B, with quantum characteristic at least three. We give an explicit description of the action of the Khovanov–Lauda–Rouquier algebra generators ψ1,…,ψn−1 on the basis elements of S((n−m),(1m)). Introducing certain Specht module homomorphisms, we construct irreducible submodules of these Specht modules, and thereby completely determine the composition series of Specht modules labelled by hook bipartitions.

Graded Skew Specht Modules and Cuspidal Modules for Khovanov-Lauda-Rouquier Algebras of Affine Type A

Kleshchev, Mathas and Ram. Proc. Lond. Math. Soc. (2) 105, 1245–1289 (2012) gave a presentation for graded Specht modules over Khovanov-Lauda-Rouquier algebras of finite and affine type A. We show that this construction can be applied more generally to skew shapes to give a presentation of graded skew Specht modules, which arise as subquotients of restrictions of Specht modules. As an application, we show that cuspidal modules associated to a balanced convex preorder in affine type A are skew Specht modules for certain hook shapes.

On the Isotypic Decomposition of Cohomology Modules of Symmetric Semi-algebraic Sets: Polynomial Bounds on Multiplicities

Abstract We consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of degrees bounded by a fixed constant d. We prove that if a Specht module, $\mathbb{S}^{\lambda }$, appears with positive multiplicity in the isotypic decomposition of the cohomology modules of such sets, then the rank of the partition $\lambda$ is bounded by O(d). This implies a polynomial (in the dimension of the ambient space) bound on the number of such modules. Furthermore, we prove a polynomial bound on the multiplicities of those that do appear with positive multiplicity in the isotypic decomposition of the abovementioned cohomology modules. We give some applications of our methods in proving lower bounds on the degrees of defining polynomials of certain symmetric semi-algebraic sets, as well as improved bounds on the Betti numbers of the images under projections of (not necessarily symmetric) bounded real algebraic sets, improving in certain situations prior results of Gabrielov, Vorobjov, and Zell.

Homomorphisms from Specht Modules to Signed Young Permutation Modules

We construct a class $\Theta_{\mathscr{R}}$ of homomorphisms from a Specht module $S_{\mathbb{Z}}^{\lambda}$ to a signed permutation module $M_{\mathbb{Z}}(\alpha|\beta)$ which generalises James's construction of homomorphisms whose codomain is a Young permutation module. We show that any $\phi \in \operatorname{Hom}_{{\mathbb{Z}}\mathfrak{S}_{n}}\big(S_{\mathbb{Z}}^\lambda, M_{\mathbb{Z}}(\alpha|\beta)\big)$ lies in the $\mathbb{Q}$-span of $\Theta_{\text{sstd}}$, a subset of $\Theta_{\mathscr{R}}$ corresponding to semistandard $\lambda$-tableaux of type $(\alpha|\beta)$. We also study the conditions for which $\Theta^{\mathbb{F}}_{\mathrm{sstd}}$ - a subset of $\operatorname{Hom}_{\mathbb{F}\mathfrak{S}_{n}}\big(S_{\mathbb{F}}^\lambda,M_{\mathbb{F}}(\alpha|\beta)\big)$ induced by $\Theta_{\mathrm{sstd}}$ - is linearly independent, and show that it is a basis for $\operatorname{Hom}_{\mathbb{F}\mathfrak{S}_{n}}\big(S_{\mathbb{F}}^\lambda,M_{\mathbb{F}}(\alpha|\beta)\big)$ when $\mathbb{F}\mathfrak{S}_{n}$ is semisimple.

Periodicity in the cohomology of symmetric groups via divided powers

A famous theorem of Nakaoka asserts that the cohomology of the symmetric group stabilizes. The first author generalized this theorem to non-trivial coefficient systems, in the form of FI-modules over a field, though one now obtains periodicity of the cohomology instead of stability. In this paper, we further refine these results. Our main theorem states that if M is a finitely generated FI-module over a noetherian ring k then ⨁ n ⩾ 0 H t ( S n , M n ) admits the structure of a D-module, where D is the divided power algebra over k in a single variable, and moreover, this D-module is ‘nearly’ finitely presented. This immediately recovers the periodicity result when k is a field, but also shows, for example, how the torsion varies with n when k = Z . Using the theory of connections on D-modules, we establish sharp bounds on the period in the case where k is a field. We apply our theory to obtain results on the modular cohomology of Specht modules and the integral cohomology of unordered configuration spaces of manifolds.

DIAGRAM ALGEBRAS, DOMINANCE TRIANGULARITY AND SKEW CELL MODULES

We present an abstract framework for the axiomatic study of diagram algebras. Algebras that fit this framework possess analogues of both the Murphy and seminormal bases of the Hecke algebras of the symmetric groups. We show that the transition matrix between these bases is dominance unitriangular. We construct analogues of the skew Specht modules in this setting. This allows us to propose a natural tableaux theoretic framework in which to study the infamous Kronecker problem.

Filters in the partition lattice

Given a filter $$\Delta $$Δ in the poset of compositions of n, we form the filter $$\Pi ^{*}_{\Delta }$$źΔź in the partition lattice. We determine all the reduced homology groups of the order complex of $$\Pi ^{*}_{\Delta }$$źΔź as $${\mathfrak S}_{n-1}$$Sn-1-modules in terms of the reduced homology groups of the simplicial complex $$\Delta $$Δ and in terms of Specht modules of border shapes. We also obtain the homotopy type of this order complex. These results generalize work of Calderbank---Hanlon---Robinson and Wachs on the d-divisible partition lattice. Our main theorem applies to a plethora of examples, including filters associated with integer knapsack partitions and filters generated by all partitions having block sizes a or b. We also obtain the reduced homology groups of the filter generated by all partitions having block sizes belonging to the arithmetic progression $$a, a + d, \ldots , a + (a-1) \cdot d$$a,a+d,ź,a+(a-1)·d, extending work of Browdy.

Cohomology of SL2 and related structures

ABSTRACTLet SL2 be an algebraic group defined over an algebraically closed field k of characteristic p > 0. In this paper, we provide a closed formula for for Weyl SL2-modules V(m) when n ≤ 2p − 3. For n > 2p − 3, an exponential bound, only depending on n, is obtained for . Analogous results are also established for the extension spaces between Weyl modules V(m1) and V(m2). As a by-product, our results and techniques give explicit upper bounds for the dimensions of cohomology of the Specht modules of symmetric groups, and the cohomology of simple modules of SL2 and the finite group of Lie type .

A note on symmetry classes of tensors and Specht modules

We construct induced bases for critical orbital spaces indexed by standard tableaux. As a corollary we obtain an explicit isomorphism of C[Sm]-modules between Specht modules and critical orbital spaces which yields a realization of the complete set of complex irreducible representations of the symmetric group in terms of decomposable symmetrized tensors.

Injective Schur modules

We determine the partitions λ for which the corresponding induced module (or Schur module in the language of Buchsbaum et al., [1]) ∇(λ) is injective in the category of polynomial modules for a general linear group over an infinite field, equivalently which Weyl modules are projective polynomial modules. Since the problem is essentially no more difficult in the quantised case we address it at this level of generality. Expressing our results in terms of the representation theory of Hecke algebras at the parameter q we determine the partitions λ for which the corresponding Specht module is a Young module, when 1+q≠0. In the classical case this problem was addressed by D. Hemmer, [10]. The nature of the set of partitions appearing in our solution gives a new formulation of Carter's condition on regular partitions. On the other hand, we note, in Remark 2.22, that the result on irreducible Weyl modules for the quantised Schur algebra Sq(n,n), [15], Theorem 5.39, given in terms of Carter partitions, may be also used to obtain the main result presented here.

Homomorphisms from an arbitrary Specht module to one corresponding to a hook

We use the presentation of the Specht modules given by Kleshchev–Mathas–Ram to derive results about Specht modules. In particular, we determine all homomorphisms from an arbitrary Specht module to a fixed Specht module corresponding to any hook partition. Along the way, we give a complete description of the action of the standard KLR generators on the hook Specht module. This work generalizes a result of James.

Quantum walled Brauer algebra: commuting families, Baxterization, and representations

For the quantum walled Brauer algebra, we construct its Specht modules and (for generic parameters of the algebra) seminormal modules. The latter construction yields the spectrum of a commuting family of Jucys–Murphy elements. We also propose a Baxterization prescription; it involves representing the quantum walled Brauer algebra in terms of morphisms in a braided monoidal category and introducing parameters into these morphisms, which allows constructing a ‘universal transfer matrix’ that generates commuting elements of the algebra.

Dyck tilings and the homogeneous Garnir relations for graded Specht modules

Suppose lambda and mu are integer partitions with lambda supseteq mu . Kenyon and Wilson have introduced the notion of a cover-inclusive Dyck tiling of the skew Young diagram lambda backslash mu , which has applications in the study of double-dimer models. We examine these tilings in more detail, giving various equivalent conditions and then proving a recurrence which we use to show that the entries of the transition matrix between two bases for a certain permutation module for the symmetric group are given by counting cover-inclusive Dyck tilings. We go on to consider the inverse of this matrix, showing that its entries are determined by what we call cover-expansive Dyck tilings. The fact that these two matrices are mutual inverses allows us to recover the main result of Kenyon and Wilson. We then discuss the connections with recent results of Kim et al., who give a simple expression for the sum, over all mu , of the number of cover-inclusive Dyck tilings of lambda backslash mu . Our results provide a new proof of this result. Finally, we show how to use our results to obtain simpler expressions for the homogeneous Garnir relations for the universal Specht modules introduced by Kleshchev, Mathas and Ram for the cyclotomic quiver Hecke algebras.

Signed Young modules and simple Specht modules

By a result of Hemmer, every simple Specht module of a finite symmetric group over a field of odd characteristic is a signed Young module. While Specht modules are parametrized by partitions, indecomposable signed Young modules are parametrized by certain pairs of partitions. The main result of this article establishes the signed Young module labels of simple Specht modules. Along the way we prove a number of results concerning indecomposable signed Young modules that are of independent interest. In particular, we determine the label of the indecomposable signed Young module obtained by tensoring a given indecomposable signed Young module with the sign representation. As consequences, we obtain the Green vertices, Green correspondents, cohomological varieties, and complexities of all simple Specht modules and a class of simple modules of symmetric groups, and extend the results of Gill on periodic Young modules to periodic indecomposable signed Young modules.