Searcharxiv⌕ Search

arXiv subjects

Samuel Creedon

Publications and source records attributed to Samuel Creedon.

9 recordsLinked to original sources
arXiv2026

Kostant's problem for permutations of shape $(n-2,1,1)$ and $(n-3,2,1)$

For a permutation $z$ in the symmetric group $\mathrm{S}_{n}$, denote by $L_{z}$ the corresponding simple highest weight module in the principal block of the BGG category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_{n}(\mathbb{C})$. In this paper, we provide a combinatorial answer to Kostant's problem for the modules $L_{z}$ when $z$ has shape (associated Young diagram/integer partition via Robinson-Schensted correspondence) equal to $(n-2,1,1)$ or $(n-3,2,1)$. Moreover, we verify that certain closely related conjectures hold for such permutations, including the Indecomposability Conjecture, which states that applying any indecomposable projective functor to the corresponding simple highest weight module outputs either an indecomposable module or zero.

math.RT↗
arXiv2026

Kostant cuspidal permutations

In relation to Kostant's problem for simple highest weight modules over the general linear Lie algebra, we prove a persistence result for Kostant negative consecutive patterns. Inspired by it, we introduce the notion of a Kostant cuspidal permutation as a minimal Kostant negative consecutive pattern. It is shown that Kostant cuspidality is an invariant of a Kazhdan-Lusztig left cell. We describe four infinite families of Kostant cuspidal involutions, including a complete classification of Kostant cuspidal fully commutative involutions. In particular, we show that the number of new Kostant cuspidal elements can be arbitrarily large, when the rank grows. This provides some potential explanation why Kostant's problem is hard.

math.RT↗
arXiv2025

Hecke combinatorics, K{\aa}hrstr{\"o}m's conditions and Kostant's problem

This paper discusses various aspects of the Hecke algebra combinatorics that are related to conditions appearing in K{\aa}hrstr{\"o}m's conjecture that addresses Kostant's problem for simple highest weight modules in the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ for the complex Lie algebra $\mathfrak{sl}_n$. In particular, we study cyclic submodules of the regular Hecke module that are generated by the elements of the (dual) Kazhdan-Lusztig basis as well as the problem of left cell invariance for both categorical and combinatorial K{\aa}hrstr{\"o}m's conditions.

math.RT↗
arXiv2025

A stability phenomenon in Kazhdan-Lusztig combinatorics

We prove that, when $n$ goes to infinity, the expression, with respect to the dual Kazhdan-Lusztig basis, of the product $\hat{\underline{H}}_x\underline{H}_y$ of elements of the dual and the usual Kazhdan-Lusztig bases in the Hecke algebra of the symmetric group $S_n$ stabilizes. As an application, we define the action of projective functors on the principal block of category $\mathcal{O}$ for $\mathfrak{sl}_\infty$ and show that the subcategory of finite length objects is stable under this action. As a bonus, we also prove that this latter block is Koszul, answering, for this block, a question from \cite{CP}.

math.RT↗
arXiv2025

Consecutive Patterns, Kostant's Problem and Type $A_6$

For a permutation $w$ in the symmetric group $\mathfrak{S}_{n}$, let $L(w)$ denote the simple highest weight module in the principal block of the BGG category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_{n}(\mathbb{C})$. We first prove that $L(w)$ is Kostant negative whenever $w$ consecutively contains certain patterns. We then provide a complete answer to Kostant's problem in type $A_{6}$ and show that the indecomposability conjecture also holds in type $A_{6}$, that is, applying an indecomposable projective functor to a simple module outputs either an indecomposable module or zero.

math.RT↗
arXiv2022

A Centraliser Analogue to the Farahat-Higman Algebra

We define a family of algebras \mathsf{FH}_{m} which generalise the Farahat-Higman algebra introduced in [FH59] by replacing the role of the center of the group algebra of the symmetric groups with centraliser algebras of symmetric groups. These algebras have a basis indexed by marked cycle shapes, combinatorial objects which generalise proper integer partitions. We analyse properties of marked cycle shapes and of the algebras \mathsf{FH}_{m}, demonstrating that some of the former govern the latter. The main theorem of the paper proves that the algebra \mathsf{FH}_{m} is isomorphic to the tensor product of the degenerate affine Hecke algebra with the algebra of symmetric functions.

math.RT↗
arXiv2021

Defining an Affine Partition Algebra

We define an affine partition algebra by generators and relations and prove a variety of basic results regarding this new algebra analogous to those of other affine diagram algebras. In particular we show that it extends the Schur-Weyl duality between the symmetric group and the partition algebra. We also relate it to the affine partition category recently defined by J. Brundan and M. Vargas. Moreover, we show that this affine partition category is a full monoidal subcategory of the Heisenberg category.

math.RT↗
arXiv2020

The Center of the Partition Algebra

In this paper we show that the center of the partition algebra $\mathcal{A}_{2k}(\delta)$, in the semisimple case, is given by the subalgebra of supersymmetric polynomials in the normalised Jucys-Murphy elements. For the non-semisimple case, such a subalgebra is shown to be central, and in particular it is large enough to recognise the block structure of $\mathcal{A}_{2k}(\delta)$. This allows one to give an alternative description for when two simple $\mathcal{A}_{2k}(\delta)$-modules belong to the same block.

math.RT↗