Braids, Shuffles and Symmetrizers

Brunnian braids和从双重shuffle李代数到Kashiwara-Vergne李代数的包含

Brunnian braids and the inclusion from double shuffle Lie algebra to Kashiwara-Vergne Lie algebra

本文证明了Brunnian braids的双重shuffle李代数通过infinitesimal hexagon方程注入到对称Kashiwara-Vergne李代数，并扩展了这一包含到两个方向：利用Brunnian Lie代数的下正则系列以及建立pentagon方程映射、stuffle coproduct、divergence映射和necklace cobracket之间的显式联系。

Quantum group covariant (anti)symmetrizers, epsilon-tensors, vielbein, Hodge map and Laplacian

GL_q(N)- and SO_q(N)-covariant deformations of the completely symmetric/antisymmetric projectors with an arbitrary number of indices are explicitly constructed as polynomials in the braid matrices. The precise relation between the completely antisymmetric projectors and the completely antisymmetric tensor is determined. Adopting the GL_q(N)- and SO_q(N)-covariant differential calculi on the corresponding quantum group covariant noncommutative spaces C_q^N, R_q^N, we introduce a generalized notion of vielbein basis (or "frame"), based on differential-operator-valued 1-forms. We then give a thor…

Generalized Matsumoto-Tits sections and quantum quasi-shuffle algebras

In this paper generalized Matsumoto-Tits sections lifting permutations to the algebra associated to a generalized virtual braid monoid are defined. They are then applied to study the defining relations of the quantum quasi-shuffle algebras via the total symmetrization operator.

A quantum shuffle approach to quantum determinants

Let $\bigwedge_σV=\bigoplus_{k\geq 0}\bigwedge_σ^kV$ be the quantum exterior algebra associated to a finite-dimensional braided vector space $(V,σ)$. For an algebra $\mathfrak{A}$, we consider the convolution product on the graded space $\bigoplus_{k\geq 0}\mathrm{Hom}\big(\bigwedge_σ^kV,\bigwedge_σ^kV\otimes \mathfrak{A}\big)$. Using this product, we define a notion of quantum minor determinant of a map from $V$ to $V\otimes \mathfrak{A}$, which coincides with the classical one in the case that $\mathfrak{A}$ is the FRT algebra corresponding to $U_q(\mathfrak{sl}_N)$. We establish quantum Lap…

Fox-Neuwirth 胞腔、量子洗牌代数与结式的特征和

Fox-Neuwirth cells, quantum shuffle algebras, and character sums of the resultant

本文利用拓扑方法计算多重穿孔平面辫群同调并证明混合辫群同调的消没范围，从而给出给定次数首一无平方因子多项式对上结式特征和的上界，回答了 Ellenberg 和 Shusterman 在二次情形下的问题。

Shuffle algebras and perverse sheaves

We relate shuffle algebras, as defined by Nichols, Feigin-Odesskii and Rosso, to perverse sheaves on symmetric products of the complex line (i.e., on the spaces of monic polynomials stratified by multiplicities of roots). More precisely, we construct an equivalence between: (i) Braided Hopf algebras of a certain type. (ii) Factorizable collections of perverse sheaves on all the symmetric products. Under this eqiuvalence, the Nichols algebra associated to an object V corresponds to the collection of the intersection cohomology extensions of the local systems on the open configuration spaces ass…

Toric braids and $(m,n)$-parking functions

The Dyck path algebra construction of Carlsson and Mellit from arXiv:1508.06239 is interpreted as a representation of "the positive part" of the group of toric braids. Then certain sums over $(m,n)$-parking functions are related to evaluations of this representation on some special braids. The compositional $(km,kn)$-shuffle conjecture of Bergeron, Garsia, Leven and Xin from arXiv:1404.4616 is then shown to be a corollary of this relation.

Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle's conjecture for function fields

The purpose of this paper is to prove the upper bound in Malle's conjecture on the distribution of finite extensions of $\mathbb{F}_q(t)$ with specified Galois group. As in previous work of Ellenberg-Venkatesh-Westerland, our result is based upon computations of the homology of braid groups with certain (exponential) coefficients. However, the approach in this paper is new, relying on a connection between the cohomology of Hurwitz spaces and the cohomology of quantum shuffle algebras.