Let $m, n\in{\mathbb N}$. In this paper we study the right permutation action of the symmetric group ${\mathfrak S}_{2n}$ on the set of all the Brauer $n$-diagrams. A new basis for the free ${\mathbb Z}$-module ${\mathfrak B}_n$ spanned by these Brauer $n$-diagrams is constructed, which yields Specht filtrations for ${\mathfrak B}_n$. For any $2m$-dimensional vector space $V$ over a field of arbitrary characteristic, we give an explicit and characteristic free description of the annihilator of the $n$-tensor space $V^{\otimes n}$ in the Brauer algebra ${\mathfrak B}_n(-2m)$. In particular, we show that it is a ${\mathfrak S}_{2n}$-submodule of ${\mathfrak B}_n(-2m)$.
Let $\H_n$ be a (degenerate or non-degenerate) Hecke algebra of type $G(\ell,1,n)$, defined over a commutative ring $R$ with one, and let $S(\bmu)$ be a Specht module for $\H_n$. This paper shows that the induced Specht module $S(\bmu)\otimes_{\H_n}\H_{n+1}$ has an explicit Specht filtration.
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…
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.
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…
In this paper we study the partial Brauer $\mathbb{C}$-algebras $\mathfrak{R}_n(δ,δ')$, where $n \in \mathbb{N}$ and $δ,δ'\in\mathbb{C}$. We show that these algebras are generically semisimple, construct the Specht modules and determine the Specht module restriction rules for the restriction $\mathfrak{R}_{n-1} \hookrightarrow \mathfrak{R}_n$. We also determine the corresponding decomposition matrix, and the Cartan decomposition matrix.
This paper investigates the tilting modules of the cyclotomic q-Schur algebras, the Young modules of the Ariki-Koike algebras, and the interconnections between them. The main tools used to understand the tilting modules are contragredient duality, and the Specht filtrations and dual Specht filtrations of certain permutation modules. Surprisingly, Weyl filtrations -- which are in general more powerful than Specht filtrations -- play only a secondary role.
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…
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.
The restriction of a (dual) Specht module to a smaller symmetric group has a filtration by (dual) Specht modules of this smaller group. In the cellular structure of the group algebra of the symmetric group, the cell modules are exactly the (dual) Specht modules. The partition algebra is a cellular algebra containing the group algebra of the symmetric group. In this article, we study the structure of the restriction of a cell module to the group algebra of a symmetric group (with smaller index) and find conditions for the restriction to possess a (dual) Specht filtration.
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.
We construct a cellular basis of the walled Brauer algebra which has similar properties as the Murphy basis of the group algebra of the symmetric group. In particular, the restriction of a cell module to a certain subalgebra can be easily described via this basis. Furthermore, the mixed tensor space possesses a filtration by cell modules -- although not by cell modules of the walled Brauer algebra itself, but by cell modules of its image in the algebra of endomorphisms of mixed tensor space.
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.
Let $V$ be a $2m$-dimensional symplectic vector space over an algebraically closed field $K$. Let $\mbb_n^{(f)}$ be the two-sided ideal of the Brauer algebra $\mbb_n(-2m)$ over $K$ generated by $e_1e_3... e_{2f-1}$, where $0\leq f\leq [n/2]$. Let $\mathcal{HT}_{f}^{\otimes n}$ be the subspace of partially harmonic tensors of valence $f$ in $V^{\otimes n}$. In this paper, we prove that $\dim\mathcal{HT}_f^{\otimes n}$ and $\dim\End_{KSp(V)}\Bigl(V^{\otimes n}/V^{\otimes n}\mbb_n^{(f)}\Bigr)$ are both independent of $K$, and the natural homomorphism from $\mbb_n(-2m)/\mbb_n^{(f)}$ to $\End_{KSp(…
Let n be a positive integer and let p be a prime. Suppose that we take a partition of n, and obtain another partition by moving a node from one row to a shorther row. Carter and Payne showed that if the p-residue of the removed and added positions is the same, then there is a non-zero homomorphism between the corresponding Specht modules for the symmetric group of degree n, defined over a field of characteristic p. In this paper we give a very simple description of such a homomorphism, as a map between polytabloids, using the action of a Murphy-Jucys element. We also present a proof that in th…
There are similarities between algebraic Lie theory and a geometric description of the blocks of the Brauer algebra in characteristic zero. Motivated by this, we study the alcove geometry of a certain reflection group action. We provide analogues of translation functors for a tower of recollement, and use these to construct Morita equivalences between blocks containing weights in the same facet. Moreover, we show that the determination of decomposition numbers for the Brauer algebra in characteristic zero can be reduced to a study of the block containing the weight 0. We define parabolic Kazhd…
In this paper we construct non-negative gradings on a basic Brauer tree algebra $A_Γ$ corresponding to an arbitrary Brauer tree $Γ$ of type $(m,e)$. We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra $A_S$, whose tree is a star with the exceptional vertex in the middle, to $A_Γ$. The grading on $A_S$ comes from the tight grading given by the radical filtration. To transfer gradings via derived equivalence we use tilting complexes constructed by taking Green's walk around $Γ$ (cf. [\ref{Zak}]). By computing endomorphism rings of these tilting complexes …
Let k be an algebraically closed field of characteristic p>0. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the general linear group over k in terms of cap diagrams under the assumption that p is bigger than the greatest hook length in the partitions involved. Then we introduce and study the rational Schur functor from a category of GL_n-modules to the category of modules for the walled Brauer algebra. As a corollary we obtain the decomposition numbers for the walled Brauer algebra when p is bigger than the greatest hook lengt…
The radical of the Brauer algebra B_f^x is known to be non-trivial when the parameter x is an integer subject to certain conditions (with respect to f). In these cases, we display a wide family of elements in the radical, which are explicitly described by means of the diagrams of the usual basis of B_f^x . The proof is by direct approach for x=0, and via classical Invariant Theory in the other cases, exploiting then the well-known representation of Brauer algebras as centralizer algebras of orthogonal or symplectic groups acting on tensor powers of their standard representation. This also give…
We prove a ``positive'' Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type~$A$. That is, in the Grothendieck group, we show that the sum of the pieces of the Jantzen filtration is equal to an explicit non-negative linear combination of modules $E^ν_{f,e}$, which are modular reductions of simple modules for closely connected Hecke algebras in characteristic zero. The coefficient of $E^ν_{f,e}$ in the sum formula is determined by the graded decomposition numbers in characteristic zero, which are known, and the characteristic of the field. As a consequence we see …
Let $K$ be an arbitrary field of characteristic not equal to 2. Let $m, n\in\N$ and $V$ an $m$ dimensional orthogonal space over $K$. There is a right action of the Brauer algebra $\bb_n(m)$ on the $n$-tensor space $V^{\otimes n}$ which centralizes the left action of the orthogonal group $O(V)$. Recently G.I. Lehrer and R.B. Zhang defined certain quasi-idempotents $E_i$ in $\bb_n(m)$ (see (\ref{keydfn})) and proved that the annihilator of $V^{\otimes n}$ in $\bb_n(m)$ is always equal to the two-sided ideal generated by $E_{[(m+1)/2]}$ if $\ch K=0$ or $\ch K>2(m+1)$. In this paper we extend thi…
Let k be an algebraically closed field of characteristic p>2. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the symplectic group over k in terms of cap-curl diagrams under the assumption that p is bigger than the greatest hook length in the largest partition involved. As a corollary we obtain the decomposition numbers for the Brauer algebra under the same assumptions. Our work combines ideas from work of Cox and De Visscher and work of Shalile with techniques from the representation theory of reductive groups.
