Zhiwei Yun
We define and study cocycles on a Coxeter group in each degree generalizing the sign function. When the Coxeter group is a Weyl group, we explain how the degree three cocycle arises naturally from geometry representation theory.
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On a stability of higher level Coxeter unipotent representations
该研究在完备离散赋值环上的连通约化群框架下，证明了q不太小且Coxeter型T、θ=1时高水平Deligne--Lusztig表示退化为r=1的情形，给出GL₂/SL₂下对应表示的维数与分解，并提出一般情形的符号猜想公式。
We obtain vanishing ranges for the mod $p$ cohomology of alternating subgroups of finite $p$-free Coxeter groups. Here a Coxeter group $W$ is $p$-free if the order of the product $st$ is prime to $p$ for every pair of Coxeter generators $s,t$ of $W$. Our result generalizes those for alternating groups formerly proved by Kleshchev-Nakano and Burichenko. As a byproduct, we obtain vanishing ranges for the twisted cohomology of finite $p$-free Coxeter groups with coefficients in the sign representations. In addition, a weak version of the main result is proved for a certain class of infinite Coxet…
Coxeter matrices and homological quadratic forms of $\boldsymbol{n}$-hereditary algebras
本文在高维 Auslander–Reiten 理论中研究 $n$-遗传代数的 Coxeter 矩阵与同调二次型，建立奇数 $n$ 时周期性、正定性与 $n$-表示有限性的等价判据。
Let $W$ be a Weyl group, and let $\CT_W$ be the complex toric variety attached to the fan of cones corresponding to the reflecting hyperplanes of $W$, and its weight lattice. The real locus $\CT_W(\R)$ is a smooth, connected, compact manifold with a $W$-action. We give a formula for the equivariant Euler characteristic of $\CT_W(\R)$ as a generalised character of $W$. In type $A_{n-1}$ for $n$ odd, one obtains a generalised character of $\Sym_n$ whose degree is (up to sign) the $n^{\text{th}}$ Euler number.
The action of a Coxeter group $W$ on the set of left cosets of a standard parabolic subgroup deforms to define a module $\mathcal{M}^J$ of the group's Iwahori-Hecke algebra $\mathcal{H}$ with a particularly simple form. Rains and Vazirani have introduced the notion of a quasiparabolic set to characterize $W$-sets for which analogous deformations exist; a motivating example is the conjugacy class of fixed point free involutions in the symmetric group. Deodhar has shown that the module $\mathcal{M}^J$ possesses a certain antilinear involution, called the bar operator, and a certain basis invaria…
An involution in a Coxeter group has an associated set of involution words, a variation on reduced words. These words are saturated chains in a partial order first considered by Richardson and Springer in their study of symmetric varieties. In the symmetric group, involution words can be enumerated in terms of tableaux using appropriate analogues of the symmetric functions introduced by Stanley to accomplish the same task for reduced words. We adapt this approach to the group of signed permutations. We show that involution words for the longest element in the Coxeter group $C_n$ are in bijecti…
$τ$-Hochschild (co)homology, the square of the Serre bimodule, and the Coxeter automorphism of the Tamarkin--Tsygan calculus
该研究关联有限维代数\(\Lambda\)的霍赫希尔德理论的两种丰富内容，即\(\tau\) - 霍赫希尔德（上）同调与考克斯特自同构\(\sigma_{\Lambda}\)。通过中山函子等方法揭示了相关复形及性质，证明两种细化横向，提出组合不变量等，还探讨了导出不变性问题。
Associated to every complex reflection group, we construct a lattice of quotients of its braid monoid-algebra, which we term nil-Hecke algebras, and which are obtained by killing all braid words that are "sufficiently long", as well as some integer power of each generator. These include usual nil-Coxeter algebras, nil-Temperley-Lieb algebras, and their variants, and lead to symmetric semigroup module categories which necessarily cannot be monoidal. Motivated by classical work of Coxeter (1957) and the Broue-Malle-Rouquier freeness conjecture [Crelle 1998], and continuing beyond work of the sec…
Shephard groups are unitary reflection groups arising as the symmetries of regular complex polytopes. For a Shephard group, we identify the representation carried by the principal ideal in the coinvariant algebra generated by the image of the product of all linear forms defining reflecting hyperplanes. This representation turns out to have many equivalent guises making it analogous to the sign representation of a finite Coxeter group. One of these guises is (up to a twist) the cohomology of the Milnor fiber for the isolated singularity at 0 in the hypersurface defined by any homogeneous invari…
Cluster realizations of Weyl groups and higher Teichmüller theory
针对可对称化Kac-Moody李代数构造含Weyl子群的加权箭图，推导丛变换显式公式，得到有限型情形的绿色序列与丛DT变换，并建立其与高阶Teichmüller空间丛结构的突变等价及群作用的一致性。
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.
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