Hecke algebras of simply-laced type with independent parameters

We study the (complex) Hecke algebra $\mathcal{H}_S(\mathbf{q})$ of a finite simply-laced Coxeter system $(W,S)$ with independent parameters $\mathbf{q} \in \left( \mathbb{C} \setminus\{\text{roots of unity}\} \right)^S$. We construct its irreducible representations and projective indecomposable representations. We obtain the quiver of this algebra and determine when it is of finite representation type. We provide decomposition formulas for induced and restricted representations between the algebra $\mathcal{H}_S(\mathbf{q})$ and the algebra $\mathcal{H}_R(\mathbf{q}|_R)$ with $R\subseteq S$. Our results demonstrate an interesting combination of the representation theory of finite Coxeter groups and their 0-Hecke algebras, including a two-sided duality between the induced and restricted representations.Comment: 20 pages; to appear in Algebraic Combinatoric

A gluing construction for polynomial invariants

We give a polynomial gluing construction of two groups $G_X\subseteq GL(\ell,\mathbb F)$ and $G_Y\subseteq GL(m,\mathbb F)$ which results in a group $G\subseteq GL(\ell+m,\mathbb F)$ whose ring of invariants is isomorphic to the tensor product of the rings of invariants of $G_X$ and $G_Y$. In particular, this result allows us to obtain many groups with polynomial rings of invariants, including all $p$-groups whose rings of invariants are polynomial over $\mathbb F_p$, and the finite subgroups of $GL(n,\mathbb F)$ defined by sparsity patterns, which generalize many known examples.Comment: 10 pages, to appear in Journal of algebr

0-Hecke algebra action on the Stanley-Reisner ring of the Boolean algebra

We define an action of the 0-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their (q,t)-analogues introduced by Bergeron and Zabrocki, and to a more general family of noncommutative symmetric functions having parameters associated with paths in binary trees introduced recently by Lascoux, Novelli, and Thibon. We also obtain multivariate quasisymmetric function identities, which specialize to results of Garsia and Gessel on generating functions of multivariate distributions of permutation statistics.Comment: Added connections with a family of noncommutative symmetric functions introduced recently by Lascoux, Novelli, and Thibo