Let $Σ_d$ 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 $H^1(Σ_d,{\rm{Sp}}(λ))$, of the Specht module ${\rm{Sp}}(λ)$ 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. Our method is to proceed by comparison with the cohomology for the general linear group $G(n)$ over $K$ and then to reduce to the calculation of ${\rm{Ext}}^1_{B(n)}(S^d E,K_λ)$, where $B(n)$ is a Borel subgroup of $G(n)$, $S^dE$ denotes the $d$th symmetric power of the natural module $E$ for $G(n)$ and $K_λ$ denotes the one dimensional $B(n)$-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 $G(n)$.
We aim to construct an element satisfying Hemmer's combinatorial criterion for $H^1(\mathfrak{S}_n, S^λ)$ to be non-vanishing. In the process, we discover an unexpected and surprising link between the combinatorial theory of integral designs and the representation theory of the symmetric groups.
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.
The author and Nakano recently proved that multiplicities in a Specht filtration of a symmetric group module are well-defined precisely when the characteristic is at least five. This result suggested the possibility of a symmetric group theory analogous to that of good filtrations and tilting modules for $GL_n(k)$. This paper is an initial attempt at such a theory. We obtain two sufficient conditions that ensure a module has a Specht filtration, and a formula for the filtration multiplicities. We then study the categories of modules that satisfy the conditions, in the process obtaining a new r…
Let $F$ be a field of characteristic $p$ at least 5. We study the Loewy structures of Specht modules in the principal block of $FΣ_{3p}$. We show that a Specht module in the block has Loewy length at most 4 and composition length at most 14. Furthermore, we classify which Specht modules have Loewy length 1, 2, 3, or 4, produce a Specht module having 14 composition factors, describe the second radical layer and the socle of the reducible Specht modules, and prove that if a Specht module corresponds to a partition that is $p$-regular and $p$-restricted then the head of the Specht module does not…
We present (with proof) a new family of decomposable Specht modules for the symmetric group in characteristic 2. These Specht modules are labelled by partitions of the form $(a,3,1^b)$, and are the first new examples found for thirty years. Our method of proof is to exhibit summands isomorphic to irreducible Specht modules, by constructing explicit homomorphisms between Specht modules.
For a Specht module S^λfor the symmetric group Σ_d, the cohomology H^i(Σ_d, S^λ) is known only in degree i=0. We give a combinatorial criterion equivalent to the nonvanishing of the degree i=1 cohomology, valid in odd characteristic. Our condition generalizes James' solution in degree zero. We apply this combinatorial description to give some computations of Specht module cohomology, together with an explicit description of the corresponding modules. Finally we suggest some general conjectures that might be particularly amenable to proof using this description.
Recently Donkin defined signed Young modules as a simultaneous generalization of Young and twisted Young modules for the symmetric group. We show that in odd characteristic, if a Specht module $S^λ$ is irreducible, then $S^λ$ is a signed Young module. Thus the set of irreducible Specht modules coincides with the set of irreducible signed Young modules. This provides evidence for our conjecture that the signed Young modules are precisely the class of indecomposable self-dual modules with Specht filtrations. The theorem is false in characteristic two.
Cohomology of Specht modules for the symmetric group can be equated in low degrees with corresponding cohomology for the Borel subgroup B of the general linear group GL_d(k), but this has never been exploited to prove new symmetric group results. Using work of Doty on the submodule structure of symmetric powers of the natural GL_d(k) module together with work of Andersen on cohomology for B and its Frobenius kernels, we prove new results about H^i(Σ_d, S^λ). We recover work of James in the case i=0. Then we prove two stability theorems, one of which is a "generic cohomology" result for Specht …
The paper presented here focuses on the classification of trivial source Specht modules. We completely classify the trivial source Specht modules labelled by hook partitions. We also classify the trivial source Specht modules labelled by two-part partitions in the odd characteristic case. Moreover, in the even characteristic case, we prove a result for the classification of the trivial source Specht modules labelled by partitions with 2-weight 2, which justifies a conjecture of [16].
We review a class of modules for the wreath product S(m) wr S(n) of two symmetric groups which are analogous to the Specht modules of the symmetric group, and prove a pair of branching rules for this family of modules. These branching rules describe the behaviour of these wreath product Specht modules under restriction to the wreath products S(m-1) wr S(n) and S(m) wr S(n-1). In particular, we see that these restrictions of wreath product Specht modules have Specht module filtrations, and we obtain combinatorial interpretations of the multiplicities in these filtrations.
We concern the VIGRE's conjecture; namely the complexity of a Specht module is the p-weight of the corresponding partition if and only if the partition is not p by p. In abelian defect case, we calculate the cohomological variety of the Specht modules. In particular, we show that the Specht modules have complexities exactly given by the p-weights of the corresponding partitions. For some p-regular partitions not more than p parts with empty p-core, we show that the Specht modules fit into the conjecture. For the partition p^p, we show that the Specht module has complexity p-1 and we study the …
We obtain alternative explicit Specht filtrations for the induced and the restricted Specht modules in the Hecke algebra of the symmetric group (defined over the ring $A=\mathbb Z[q^{1/2},q^{-1/2}]$ where $q$ is an indeterminate) using $C$-bases for these modules. Moreover, we provide a link between a certain $C$-basis for the induced Specht module and the notion of pairs of partitions.
In this paper we study the vertices of indecomposable Specht modules for symmetric groups. For any given indecomposable non-projective Specht module, the main theorem of the article describes a family of p-subgroups contained in its vertex. The theorem generalizes and improves an earlier result due to Wildon in "Vertices of Specht modules and blocks of the symmetric group", J. Alg 323 (2010) 2243--2256.
Using purely combinatorial methods we calculate the first degree cohomology of Specht modules indexed by two part partitions over fields of characteristic $p\ge 3$. These combinatorial methods also allow us to obtain an explicit description of all of the non-split extensions of the Specht module, $S^λ$, by the trivial module. Applying this work to partitions with more than two parts we are able to give an entirely combinatorial proof of the bound on the dimension of the first degree cohomology given by work of Donkin and Geranios. We also obtain as a corollary a result of Weber giving a far re…
We study the decomposability of Specht modules labelled by bihooks, bipartitions with a hook in each component, for the Iwahori--Hecke algebra of type $B$. In all characteristics, we determine a large family of decomposable Specht modules, and conjecture that these provide a complete list of decomposable Specht modules indexed by bihooks. We prove the conjecture for small $n$.
We define a notion of virtual Specht stability which is a relaxation of the Church-Farb notion of representation stability for sequences of symmetric group representations. Using a structural result of Nagpal, we show that $FI$-modules over fields of positive characteristic exhibit virtual Specht stability.
The reducibility of the Specht modules for the Iwahori--Hecke algebras in type $A$ is still open in the case where the defining parameter $q$ equals -1. We prove the reducibility of a large class of Specht modules for these algebras.
The reducible Specht modules for the Hecke algebra $\mathcal{H}_{\mathbb{F},q}(\mathfrak{S}_n)$ have been classified except when $q=-1$. We prove one half of a conjecture which we believe classifies the reducible Specht modules when $q=-1$ and $\mathbb{F}$ has characteristic 0.
In this article, we consider indecomposable Specht modules with abelian vertices. We show that the corresponding partitions are necessarily $p^2$-cores where $p$ is the characteristic of the underlying field. Furthermore, in the case of $p\geq 3$, or $p=2$ and $μ$ is 2-regular, we show that the complexity of the Specht module $S^μ$ is precisely the $p$-weight of the partition $μ$. In the latter case, we classify Specht modules with abelian vertices. For some applications of the above results, we extend a result of M. Wildon and compute the vertices of the Specht module $S^{(p^p)}$ for $p\geq 3…
Brundan, Kleshchev and Wang equip the Specht modules $S_λ$ over the cyclotomic Khovanov--Lauda--Rouquier algebra $\mathscr{H}_n^Λ$ with a homogeneous $\mathbb{Z}$-graded basis. In this paper we begin the study of graded Specht modules labelled by hook bipartitions $((n-m),(1^m))$ in level $2$ of $\mathscr{H}_n^Λ$, 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 $ψ_1,\dots,ψ_{n-1}$ on the basis elements of $S_{((n-m),(1^m))}$. Introducing certain Specht…
We give a decomposition as a direct sum of indecomposable modules of several types of Specht modules in characteristic $2$. These include the Specht modules labelled by hooks, whose decomposability was considered by Murphy. Since the main arguments are essentially no more difficult for Hecke algebras at parameter $q=-1$, we proceed in this level of generality.
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 tensor…
In [14] Hemmer conjectures that the module of fixed points for the symmetric group $Σ_m$ of a Specht module for $Σ_n$ (with $n>m$), over a field of positive characteristic $p$, has a Specht series, when viewed as a $Σ_{n-m}$-module. We provide a counterexample for each prime $p$. The examples have the same form for $p\geq 5$ and we treat the cases $p=3$ and $p=2$ separately.
Let w be the p-weight of a hook partition μand E be an elementary abelian p-subgroup generated by w disjoint p-cycles. We compute the generic Jordan type of the Specht module corresponding to the partition μrestricted to E. In particular, we show that the complexity of the Specht module is precisely the p-weight of the partition μ.
We suggest a simple definition for categorification of modules over rings and illustrate it by categorifying integral Specht modules over the symmetric group and its Hecke algebra via the action of translation functors on some subcategories of category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_n(\mathbb{C})$.
The construction of all irreducible modules of the symmetric groups over an arbitrary field which reduce to Specht modules in the case of fields of characteristic zero is given by G.D.James. Halıcıoğlu and Morris describe a possible extension of James' work for Weyl groups in general, where Young tableaux are interpreted in terms of root systems. In this paper we show how to construct submodules of Specht modules for Weyl groups.
We give a family of pairs of Weyl modules for which the corresponding homomorphism space is at least 2-dimensional. Using this result we show that for fixed parameters $e>0$ and $p\geq 0$ there exist arbitrarily large homomorphism spaces between pairs of Weyl modules.
In this paper we generalize a theorem due to Lyle, extending its application to the setting of the Ariki-Koike algebra, and in doing so establish an analogue of the kernel intersection theorem. This in turn provides us with a means towards constructing homomorphisms between Specht modules for the Ariki-Koike algebra.
Recently, the first two authors have defined a Z-grading on group algebras of symmetric groups and more generally on the cyclotomic Hecke algebras of type G(l,1,d). In this paper we explain how to grade Specht modules over these algebras.
Let n geq 1, let lambda be a partition of n, let mu be a partition arising from lambda by a downwards shift of two boxes situated at the bottom of a column. We give a formula for a ZS_n-linear morphism of order m between the corresponding Specht modules over Z/(m), where m is the box shift length (divided by two in certain combinatorially specified cases). Reformulated, this yields an extension of the corresponding Specht modules over Z of order m in Ext^1.
This paper studies the vertices, in the sense defined by J. A. Green, of Specht modules for symmetric groups. The main theorem gives, for each indecomposable non-projective Specht module, a large subgroup contained in one of its vertices. A corollary of this theorem is a new way to determine the defect groups of symmetric groups. We also use it to find the Green correspondents of a particular family of simple Specht modules; as a corollary, we get a new proof of the Brauer correspondence for blocks of the symmetric group. The proof of the main theorem uses the Brauer homomorphism on modules, a…
Let $F$ be a field of characteristic $p$. We show that $\Hom_{FΣ_n}(S^λ, S^μ)$ can have arbitrarily large dimension as $n$ and $p$ grow, where $S^λ$ and $S^μ$ are Specht modules for the symmetric group $Σ_n$. Similar results hold for the Weyl modules of the general linear group. Every previously computed example has been at most one-dimensional, with the exception of Specht modules over a field of characteristic two. The proof uses the work of Chuang and Tan, providing detailed information about the radical series of Weyl modules in Rouquier blocks.
We consider the problem of classifying irreducible Specht modules for the Iwahori-Hecke algebra of type B with parameters Q,q. We solve this problem completely in the case where q is not a root of unity, and in the case q=-1 we reduce the problem to the corresponding problem in type A.
We continue the study of Specht modules labelled by hook bipartitions for the Iwahori--Hecke algebra of type $B$ with $e\in\{3,4,\dots\}$ via the cyclotomic Khovanov--Lauda--Rouquier algebra $\mathscr{H}_n^Λ$. Over an arbitrary field, we explicitly determine the graded decomposition submatrices for $\mathscr{H}_n^Λ$ comprising rows corresponding to hook bipartitions.
Dipper, James and Murphy generalized the classical Specht module theory to Hecke algebras of type $B_n$. On the other hand, for any choice of a monomial order on the parameters in type $B_n$, we obtain corresponding Kazhdan--Lusztig cell modules. In this paper, we show that the Specht modules are naturally equivalent to the Kazhdan--Lusztig cell modules {\em if} we choose the dominance order on the parameters, as in the ``asymptotic case'' studied by Bonnafé and the second named author. We also give examples which show that such an equivalence does not hold for other choices of monomial orders…
Let $p$ be a prime and $\mathbb{F}$ a field of characteristic $p$, and let $\mathcal{H}_n$ denote the Iwahori--Hecke algebra of the symmetric group $\mathfrak{S}_n$ over $\mathbb{F}$ at $q=-1$. We prove that there are only finitely many partitions $λ$ such that both $λ$ and $λ'$ are 2-singular and the Specht module $S^λ$ for $\mathcal{H}_{|\la|}$ is irreducible.
Let $S_λ$ denote the Specht module defined by Dipper and James for the Iwahori-Hecke algebra $\mathscr{H}_n$ of the symmetric group $\mathfrak{S}_n$. When $e=2$ we determine the decomposability of all Specht modules corresponding to hook partitions $(a,1^b)$. We do so by utilising the Brundan-Kleshchev isomorphism between $\mathscr{H}$ and a Khovanov-Lauda-Rouquier algebra and working with the relevant KLR algebra, using the set-up of Kleshchev-Mathas-Ram. When $n$ is even, we easily arrive at the conclusion that $S_λ$ is indecomposable. When $n$ is odd, we find an endomorphism of $S_λ$ and us…
Let n be a positive integer and let Sigma_n be the symmetric group of degree n. Let S^lambda be the Specht module for Sigma_n corresponding to a partition lambda of n, defined over a field F of odd characteristic. We find the indecomposable components of the restriction of S^lambda to Sigma_{n-1}, and of the induction of S^lambda to Sigma_{n+1}. Namely, if b and B are block idempotents of FSigma_{n-1} and FSigma_{n+1} respectively, then the modules S^lambda b and S^lambda B are 0 or indecomposable. We give examples to show that the assumption that F has odd characteristic cannot be dropped.
Over fields of characteristic zero, there are well known constructions of the irreducible representations, due to A Young, and of irreducible modules, called Specht modules, due to W Specht, for the symmetric groups $S_{n}$ which are based on elegant combinatorial concepts connected with Young tableaux etc.(see, e.g.{\bf [13]}). James {\bf [12]} extended these ideas to construct irreducible representations and modules over arbitrary field. Al-Aamily, Morris and Peel {\bf [1]} showed how this construction could be extended cover the Weyl groups of type $B_{n}$. In {\bf [14]} Morris described a …
A construction of bases for cell modules of the Birman--Murakami--Wenzl (or B--M--W) algebra $B_n(q,r)$ by lifting bases for cell modules of $B_{n-1}(q,r)$ is given. By iterating this procedure, we produce cellular bases for B--M--W algebras on which a large abelian subalgebra, generated by elements which generalise the Jucys--Murphy elements from the representation theory of the Iwahori--Hecke algebra of the symmetric group, acts triangularly. The triangular action of this abelian subalgebra is used to provide explicit criteria, in terms of the defining parameters $q$ and $r$, for B--M--W alg…
Over fields of characteristic zero, there are well known construction of the irreducible representations and of irreducible modules, called Specht modules for the symmetric groups $S_{n}$ which are based on elegant combinatorial concepts connected with Young tableaux etc.(see, e.g.[10]). James [8] extended these ideas to construct irreducible representations and modules over arbitrary field. Al-Aamily, Morris and Peel [ 1 ] showed how this construction could be extended to deal with the Weyl groups of type $B_{n}$. In [11] the second author described a possible extension of James' work for Wey…
We construct explicit resolutions of Weyl modules by divided powers and of co-Specht modules by permutational modules. We also prove a conjecture of Boltje-Hartmann on resolutions of co-Specht modules.
Kleshchev, Mathas and Ram (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.
Let $n$ be a positive integer, and let $\mathscr{H}_n$ denote the affine KLR algebra in type A. Kleshchev, Mathas and Ram have given a homogeneous presentation for graded column Specht modules $\operatorname{S}_λ$ for $\mathscr{H}_n$. Given two multipartitions $λ$ and $μ$, we define the notion of a \emph{dominated} homomorphism $\operatorname{S}_λ\to\operatorname{S}_μ$, and use the KMR presentation to prove a generalised column removal theorem for graded dominated homomorphisms between Specht modules. In the process, we prove some useful properties of $\mathscr{H}_n$-homomorphisms between Spec…
This paper proves that the restriction of a Specht module for a (degenerate or non-degenerate) cyclotomic Hecke algebra, or KLR algebra, of type A has a Specht filtration.
The elementary divisors of the Gram matrices of Specht modules S^lambda over the symmetric group are determined for two-row partitions and for two-column partitions lambda. More precisely, the subquotients of the Jantzen filtration are calculated using Schaper's formula. Moreover, considering a general partition lambda of n at a prime p > n - lambda_1, the only possible non trivial composition factor of S_p^lambda is induced by the morphism of Carter and Payne, as shown by means of Kleshchev's modular branching rule. This enables the Jantzen filtration to be calculated in this case as well.
We construct a class $Θ_{\mathscr{R}}$ of homomorphisms from a Specht module $S_{\mathbb{Z}}^λ$ to a signed permutation module $M_{\mathbb{Z}}(α|β)$ which generalises James's construction of homomorphisms whose codomain is a Young permutation module. We show that any $ϕ\in \operatorname{Hom}_{\mathbb{Z}\mathfrak{S}_{n}}\big(S_{\mathbb{Z}}^λ, M_{\mathbb{Z}}(α|β)\big)$ lies in the $\mathbb{Q}$-span of $Θ_{\text{sstd}}$, a subset of $Θ_{\mathscr{R}}$ corresponding to semistandard $λ$-tableaux of type $(α|β)$. We also study the conditions for which $Θ^{\mathbb{F}}_{\mathrm{sstd}}$ - a subset of $\…
In a previous paper, the authors studied the radical filtration of a Weyl module $Δ_ζ(λ)$ for quantum enveloping algebras $U_ζ(\overset\circ{\mathfrak g})$ associated to a finite dimensional complex semisimple Lie algebra $\overset\circ{\mathfrak g}$. There $ζ^2=\sqrt[e]{1}$ and $λ$ was, initially, required to be $e$-regular. Some additional restrictions on $e$ were required---e.g., $e>h$, the Coxeter number, and $e$ odd. Translation to a facet gave an explicit semisimple series for all quantum Weyl modules with singular, as well as regular, weights. That is, the sections of the filtration are…
We give a combinatorial description for when the Specht module of an arbitrary diagram admits a (complete) branching rule. This description, given in terms of the maximal rectangles of the diagram, generalizes all previously known branching rules for Specht modules, such as those given by Reiner and Shimozono for northwest diagrams and by the present author for forest diagrams.
For any prime p, we construct, and simultaneously count, all of the complex Specht modules in a given p-block of the symmetric group which remain irreducible when reduced modulo p. We call the Specht modules with this property p-irreducible modules. Recently Fayers has proven a conjecture of James and Mathas that provides a characterization of the partitions that correspond to the p-irreducible modules. In this paper we present a method for decomposing the partitions corresponding to p-irreducible modules, and we use this decomposition to construct and count all of the partitions corresponding…
We prove that the dimension of the Specht module of a forest $G$ is the same as the normalized volume of the matching polytope of $G$. We also associate to $G$ a symmetric function $s_G$ (analogous to the Schur symmetric function $s_λ$ for a partition $λ$) and investigate its combinatorial and representation-theoretic properties in relation to the Specht module and Schur module of $G$. We then use this to define notions of standard and semistandard tableaux for forests.
