Related papers: Affine twisted length function

Let $W$ be a finite Weyl group and $\widetilde W$ the corresponding affine Weyl group. A random element of $\widetilde W$ can be obtained as a reduced random walk on the alcoves of $\widetilde W$. By a theorem of Lam (Ann. Prob. 2015), such…

In arxiv:1602.04254, we have defined polynomial Witt vectors functor from vector spaces over a perfect field $k$ of positive characteristic $p$ to abelian groups. In this paper, we use polynomial Witt vectors to construct a functorial…

The analytic properties of the standard twist $F(s,\alpha)$, where $F(s)$ belongs to a wide class of $L$-functions, are of prime importance in describing the structure of the Selberg class. In this paper we present a deeper study of such…

Let $A$ be either a simplicial complex $K$ or a small category $\mathcal C$ with $V(A)$ as its set of vertices or objects. We define a twisted structure on $A$ with coefficients in a simplicial group $G$ as a function $$ \delta\colon…

Determining the length of short conjugators in a group can be considered as an effective version of the conjugacy problem. The conjugacy length function provides a measure for these lengths. We study the behaviour of conjugacy length…

We define the twisted affine Yangian of type $C$ and construct surjective homomorphisms from twisted affine Yangians of type $C$ to the universal enveloping algebra of the rectangular $W$-algebra associated with $\mathfrak{so}(ln)$ and a…

The limit weak order on an affine Weyl group was introduced by Lam and Pylyavskyy in their study of total positivity for loop groups. They showed that in the case of the affine symmetric group the minimal elements of this poset coincide…

We introduce functional Wulff shapes based on the classical construction for compact convex sets. With this new tool, we establish a functional version of Aleksandrov's variational lemma in the family of convex functions with compact…

A twist property is developed which imparts certain properties on the twisted group algebra. These include an involution * satisfying (xy)*=y*x* and an inner product satisfying <xy,z> = <x,zy*> and <xy,z>=<y,x*z>. Examples of twisted group…

We compute the finite-dimensional Nichols algebras over the sum of two simple Yetter-Drinfeld modules V and W over non-abelian quotients of a certain central extension of the dihedral group of eight elements or SL(2,3), and such that the…

Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, formulate the intertwiners, and describe the…

We show that a specialization in Weyl character formula can be carried out in such a way that its right-hand side becomes simply a Schur Function. For this, we need the use of fundamental weights. In the generic definition, an Elementary…

Let $\mathcal{A}$ and $\mathcal{B}$ be two alternative $W^{*}$-factors. In this paper, we proved that a bijective mapping $\Phi :\mathcal{A}\rightarrow \mathcal{B}$ satisfies $\Phi (ab+ba^{*})=\Phi (a)\Phi (b)+\Phi (b)\Phi (a)^{*}$ (resp.,…

We fix a monic polynomial $f(x) \in \mathbb F_q[x]$ over a finite field of characteristic $p$ of degree relatively prime to $p$. Let $a\mapsto \omega(a)$ be the Teichm\"uller lift of $\mathbb F_q$, and let $\chi:\mathbb{Z}\to \mathbb…

In the context of the Springer correspondence, the Weyl group action on the Springer sheaf can be defined in two ways: via restriction or the Fourier transform. It is well-known that these two actions differ by the sign character. This was…

We construct a well-behaved Weil-\'etale complex for a large class of $\mathbb{Z}$-constructible sheaves on a regular irreducible scheme $U$ of finite type over $\mathbb{Z}$ and of dimension $1$. We then give a formula for the special value…

We propose 4-point S-matrices for three-dimensional F-theory. We will use the twistor formalism to facilitate constructing the amplitude. We write the amplitude in a way such that the F-symmetry (U-duality symmetry) is manifest. The…

We calculate normalization factors and reflection amplitudes in the W-invariant conformal quantum field theories. Using these CFT data we derive vacuum expectation values of exponential fields in affine Toda theories and related perturbed…

For a finite acyclic quiver $Q$ and the corresponding preprojective algebra $\Pi$, we study the factor algebra $\Pi_w$ associated with a element $w$ in the Coxeter group introduced by Buan-Iyama-Reiten-Scott. The algebra $\Pi_w$ has a…

In any Coxeter group, the conjugates of elements in its Coxeter generating set are called reflections and the reflection length of an element is its length with respect to this expanded generating set. In this article we give a simple…