Total nonnegativity and induced sign characters of the Hecke algebra

On combinatorial formulas for the characters of Hecke algebras

Hecke algebras are beautiful q-extensions of Coxeter groups. In this paper, we prove several results on their characters, with an emphasis on characters induced from trivial and sign representations of parabolic subalgebras. While most of the results in type A are known, our proofs are of a combinatorial nature, and generalize to (partial) results in types B and C. We also present complete descriptions of such characters for type I.

Signatures of representations of Hecke algebras and rational Cherednik algebras

Determining whether an irreducible representation of a group (or $*$-algebra) admits a non-degenerate invariant, positive-definite Hermitian form is an important problem in representation theory. In this paper, we study a related notion: that of signatures. We study representations $S^λ(q)$ of $\mathcal{H}_{n}(q)$, the Hecke algebra of type $A$ ($|q| = 1$), and representations $M_{c}(λ)$ of $\mathbb{H}_{c}$, the rational Cherednik algebra of type $A$ ($c \in \mathbb{R}$), which have unique (up to scaling) invariant Hermitian forms (here $λ$ is a partition of $n$). The signature is the number o…

Bases of the quantum matrix bialgebra and induced sign characters of the Hecke algebra

We combinatorially describe entries of the transition matrices which relate monomial bases of the zero-weight space of the quantum matrix bialgebra. This description leads to a combinatorial rule for evaluating induced sign characters of the type $A$ Hecke algebra $H_n(q)$ at all elements of the form $(1 + T_{s_{i_1}}) \cdots (1 + T_{s_{i_m}})$, including the Kazhdan-Lusztig basis elements indexed by $321$-hexagon-avoiding permutations. This result is the first subtraction-free rule for evaluating all elements of a basis of the $H_n(q)$-trace space at all elements of a basis of $H_n(q)$.

Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements

For irreducible characters $\{ χ_q^λ\,|\, λ\vdash n \}$, induced sign characters $\{ ε_q^λ\,|\, λ\vdash n \}$, and induced trivial characters $\{ η_q^λ\,|\, λ\vdash n \}$ of the Hecke algebra $H_n(q)$, and Kazhdan-Lusztig basis elements $C'_w(q)$ with $w$ avoiding the patterns 3412 and 4231, we combinatorially interpret the polynomials $χ_q^λ(q^{l(w)/2}C'_w(q))$, $ε_q^λ(q^{l(w)/2} C'_w(q))$, and $\smash{η_q^λ(q^{l(w)/2} C'_w(q))}$. This gives a new algebraic interpretation of chromatic quasisymmetric functions of Shareshian and Wachs, and a new combinatorial interpretation of special cases of …

C型$K$-Stanley对称函数与Kraśkiewicz-Hecke插入

Type C $K$-Stanley symmetric functions and Kraśkiewicz-Hecke insertion

该研究引入Kraśkiewicz插入的$K$理论版本，研究C型$K$-Stanley对称函数的GQ展开问题，证明了Lewis和Marberg关于矩形GQ函数的猜想，并给出斜GQ展开及GQ乘积的相关猜想。

$W$-graph versions of tensoring with the $§_n$ defining representation

We further develop the theory of inducing $W$-graphs worked out by Howlett and Yin in \cite{HY1}, \cite{HY2}, focusing on the case $W = §_n$. Our main application is to give two $W$-graph versions of tensoring with the $§_n$ defining representation $V$, one being $\H \tsr_{\H_J} -$ for $\H, \H_J$ the Hecke algebras of $§_n, §_{n-1}$ and the other $(\pH \tsr_{\H} -)_1$, where $\pH$ is a subalgebra of the extended affine Hecke algebra and the subscript signifies taking the degree 1 part. We look at the corresponding $W$-graph versions of the projection $V \tsr V \tsr - \to S^2 V \tsr -$. This do…