arXiv subjects
Publications and source records attributed to Sinead Lyle.
We consider the core blocks corresponding to the Hecke algebras of type B over a field of arbitrary characteristic. To each core block B, we associate two non-negative integers which determine the indexing of the Specht modules and simple modules in the block, the weight of the block, the multicharge of the algebra (up to a shift) and the block decomposition matrix.
Let $\mathcal{H}$ denote an Ariki-Koike algebra over a field of characteristic $p\geq 0$. For each $r$-multipartition ${\bf \lambda}$ of $n$, we define a $\mathcal{H}$-module $S^{{\bf \lambda}}$ and for each Kleshchev $r$-multipartition ${\bf \mu}$ of $n$, we define an irreducible $\mathcal{H}$-module $D^{{\bf \mu}}$. Given a multipartition ${\bf \lambda}$ and a Kleshchev multipartition ${\bf \mu}$ both lying in a Rouquier block and which have a common multicore, we give a closed formula for the graded decomposition number $[S^{{\bf \lambda}}:D^{{\bf \mu}}]_v$ when $p=0$.
The Rouquier blocks, also known as the RoCK blocks, are important blocks of the symmetric groups algebras and the Hecke algebras of type A, with the partitions labelling the Specht modules that belong to these blocks having a particular abacus configuration. We generalise the definition of Rouquier blocks to the Ariki-Koike algebras, where the Specht modules are indexed by multipartitions, and explore the properties of these blocks
We consider simple modules for a Hecke algebra with a parameter of quantum characteristic $e$. Equivalently, we consider simple modules $D^λ$, labelled by $e$-restricted partitions $λ$ of $n$, for a cyclotomic KLR algebra $R_n^{Λ_0}$ over a field of characteristic $p\ge 0$, with mild restrictions on $p$. If all parts of $λ$ are at most $2$, we identify a set $\mathsf{DStd}_{e,p}(λ)$ of standard $λ$-tableaux, which is defined combinatorially and naturally labels a basis of $D^λ$. In particular, we prove that the $q$-character of $D^λ$ can be described in terms of $\mathsf{DStd}_{e,p}(λ)$. We show that a certain natural approach to constructing a basis of an arbitrary $D^λ$ does not work in general, giving a counterexample to a conjecture of Mathas.
We present some blocks of Ariki-Koike algebras $\mathcal{H}_{n,r}$ for which the decomposition matrices are independent of the characteristic of the underlying field. We complete the description of the graded decomposition numbers for blocks of Ariki-Koike algebras of weight at most two, which consists of analysing the indecomposable core blocks at level $r = 3$, and give a closed formula for the decomposition numbers in this case.
We construct a new family of homomorphisms between (graded) Specht modules of the quiver Hecke algebras of type A. These maps have many similarities with the homomorphisms constructed by Carter and Payne in the special case of the symmetric groups, although the maps that we obtain are both more and less general than these.
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.
We study the homomorphism spaces between Specht modules for the Hecke algebras $\h$ of type $A$. We prove a cellular analogue of the kernel intersection theorem and a $q$-analogue of a theorem of Fayers and Martin and apply these results to give an algorithm which computes the homomorphism spaces $\Hom_{\h}(S^μ,S^λ)$ for certain pairs of partitions $λ$ and $μ$. We give an explicit description of the homomorphism spaces $\Hom_\h(S^μ,S^λ)$ where $\h$ is an algebra over the complex numbers, $λ=(λ_1,λ_2)$ and $μ$ is an arbitrary partition with $μ_1 \geq λ_2$.
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.
We prove a q-analogue of the Carter-Payne theorem in the case where the differences between the parts of the partitions are sufficiently large. We identify a layer of the Jantzen filtration which contains the image of these Carter-Payne homomorphisms and we show how these homomorphisms compose.
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.
This paper classifies the blocks of the cyclotomic Hecke algebras of type G(r,1,n) over an arbitrary field. Rather than working with the Hecke algebras directly we work instead with the cyclotomic Schur algebras. The advantage of these algebras is that the cyclotomic Jantzen sum formula gives an easy combinatorial characterization of the blocks of the cyclotomic Schur algebras. We obtain an explicit description of the blocks by analyzing the combinatorics of `Jantzen equivalence'. We remark that a proof of the classification of the blocks of the cyclotomic Hecke algebras was announced in 1999. Unfortunately, Cox has discovered that this previous proof is incomplete.
We prove a $q$-analogue of the Carter-Payne theorem for the two special cases corresponding to moving an arbitrary number of nodes between adjacent rows, or moving one node between an arbitrary number of rows. As a consequence, we show that these homomorphism spaces are one dimensional when $q \neq -1$. We apply these results to complete the classification of the reducible Specht modules for the Hecke algebras of the symmetric groups when $q \neq-1$. Our methods can also be used to determine certain other pairs of Specht modules between which there is a homomorphism. In particular, we describe the homomorphism space from the trivial module to an arbitrary Specht module.