Fused permutations algebras and degenerate affine Hecke algebras
Yoann Demesmay
TL;DR
The paper provides an explicit algebraic presentation of the fused permutations algebra $H_{k,n}$ by relating it to the cyclotomic degenerate affine Hecke algebra $\hat{\mathcal{H}}_n^{(\kappa_1,\kappa_2)}$ and its central idempotents. It constructs a complete basis for $H_{k,n}$ via signed permutations with pattern avoidance and shows that, for $n\le k$, $H_{k,n}$ is isomorphic to the quotient $\mathcal{A}_n^{(0,k+1),(k)}$ of $\hat{\mathcal{H}}_n^{(0,k+1)}$, with a concrete surjective map from the Hecke algebra to $H_{k,n}$. The representation theory is developed through the Artin–Wedderburn framework, detailing the bipartite-indexed irreducibles and the role of idempotents $F_n^{(\alpha,\beta)}$; special attention is given to the case $(\kappa_1,\kappa_2)=(0,k+1)$ and to pattern-avoiding bases in $B_n(\bar{1}\bar{2})$. For $n>k$, a refined quotient $\mathcal{A}_n^{\kappa,(k)}$ is analyzed, with bases in $B_n(\bar{1}\bar{2})$ avoiding further words, and dimension comparisons with $H_{k,n}$ highlighting regimes of equality and strict inequality. Overall, the work yields a rigorous, generator-and-relations presentation plus a combinatorial basis for $H_{k,n}$ and clarifies its precise relationship to cyclotomic degenerate affine Hecke algebras, advancing understanding of centralisers in GL$_N$ contexts.
Abstract
This paper gives an algebraic presentation of an algebra called the fused permutations algebra in the one-boundary case. It is obtained through a detailed study of the degenerate cyclotomic Hecke algebra. In particular, we prove that the fused permutations algebra is a quotient of the degenerate cyclotomic affine Hecke algebra, and we also describe a basis combinatorially in terms of signed permutations with avoiding patterns. In order to understand this quotient, we study the primitive idempotents of this degenerate cyclotomic affine Hecke algebra.