On representations of cyclotomic Hecke algebras

Induced representations and traces for chains of affine and cyclotomic Hecke algebras

Properties of relative traces and symmetrizing forms on chains of cyclotomic and affine Hecke algebras are studied. The study relies on a use of bases of these algebras which generalize a normal form for elements of the complex reflection groups $G(m,1,n)$, $m=1,2,\dots,\infty$, constructed by a recursive use of the Coxeter--Todd algorithm. Formulas for inducing, from representations of an algebra in the chain, representations of the next member of the chain are presented.

融合排列代数与退化仿射Hecke代数

Fused permutations algebras and degenerate affine Hecke algebras

本文通过研究退化 cyclotomic 仿射 Hecke 代数，证明了融合排列代数是其商代数，并给出了基于带符号排列的基底描述。

Cyclotomic shuffles

Analogues of 1-shuffle elements for complex reflection groups of type $G(m,1,n)$ are introduced. A geometric interpretation for $G(m,1,n)$ in terms of rotational permutations of polygonal cards is given. We compute the eigenvalues, and their multiplicities, of the 1-shuffle element in the algebra of the group $G(m,1,n)$. Considering shuffling as a random walk on the group $G(m,1,n)$, we estimate the rate of convergence to randomness of the corresponding Markov chain. We report on the spectrum of the 1-shuffle analogue in the cyclotomic Hecke algebra $H(m,1,n)$ for $m=2$ and small $n$.

Algebraic independence of sequences generated by (cyclotomic) harmonic sums

An expression in terms of (cyclotomic) harmonic sums can be simplified by the quasi-shuffle algebra in terms of the so-called basis sums. By construction, these sums are algebraically independent within the quasi-shuffle algebra. In this article we show that the basis sums can be represented within a tower of difference ring extensions where the constants remain unchanged. This property enables one to embed this difference ring for the (cyclotomic) harmonic sums into the ring of sequences. This construction implies that the sequences produced by the basis sums are algebraically independent ove…