Related papers: On the combinatorics of string polytopes

Let $\mathfrak{p} \subset V$ be a polytope and $\xi \in V_{\mathbb{C}}^*$. We obtain an expression for $I(\mathfrak{p}; \alpha) := \int_{\mathfrak{p}} e^{\langle \alpha, x \rangle} dx$ as a sum of meromorphic functions in $\alpha \in…

This work is part of a project on weight bases for the irreducible representations of semisimple Lie algebras with respect to which the representation matrices of the Chevalley generators are given by explicit formulas. In the case of sl_n,…

Strongly $\mathbb{Z}$-graded algebras or principal circle bundles and associated line bundles or invertible bimodules over a class of generalized Weyl algebras $\mathcal{B}(p;q, 0)$ (over a ring of polynomials in one variable) are…

We discuss aspects of topological B-type D-branes in the framework of the derived category of coherent sheaves on a Calabi-Yau 3-fold X. We analyze the link between massless D-branes and monodromies in the CFT moduli space. A classification…

We define a very general class of CHL-models associated with any string theory (bosonic or supersymmetric) compactified on an internal CFT C x T^d. We take the orbifold by a pair (g,\delta), where g is a (possibly non-geometric) symmetry of…

Regular semisimple Hessenberg varieties are subvarieties of the flag variety $\mathrm{Flag}(\mathbb{C}^n)$ arising naturally in the intersection of geometry, representation theory, and combinatorics. Recent results of…

Let Q be a finite quiver without oriented cycles, let \Lambda be the associated preprojective algebra, let g be the associated Kac-Moody Lie algebra with Weyl group W, and let n be the positive part of g. For each Weyl group element w, a…

We consider the periplectic supergroup ${\bf P} (n)$ over a ground field $\Bbbk$ of characteristic $p>2$. We show that there are four blocks of ${\bf P} (n)$ of simple supermodules $L^{\epsilon}(\lambda)$ corresponding to dominant weights…

We present a combinatorial monomial basis (or, more precisely, a family of monomial bases) in every finite-dimensional irreducible $\mathfrak{so}_{2n+1}$-module. These bases are in many ways similar to the FFLV bases for types $A$ and $C$.…

We study the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties. We associate to each pseudoline arrangement two polyhedral cones, defined in a dual manner. We prove that one of them…

We study integrals of the form \begin{equation*} \int_{-1}^1(C_n^{(\lambda)}(x))^2(1-x)^\alpha (1+x)^\beta\, dx, \end{equation*} where $C_n^{(\lambda)}$ denotes the Gegenbauer-polynomial of index $\lambda>0$ and $\alpha,\beta>-1$. We give…

Let ${\cal O}_{*}$ be the C$^{*}$-algebra defined as the direct sum of all Cuntz algebras. Then ${\cal O}_{*}$ has a non-cocommutative comultiplication $\Delta_{\phi}$ and a counit $\epsilon$. Let ${\rm BI}({\cal O}_{*})$ denote the set of…

We completely determine the minimal polynomial of an arbitrary simple highest weight module $L(\lambda)$ over a complex classical Lie algebra $\mathfrak{g}\subseteq\mathfrak{gl}_N$ relative to its defining module $\pi=\mathbb{C}^{N}$. These…

Let $W_n^+$ be the Lie algebra of the Lie algebra of vector fields on $\C^n$. In this paper, we classify all simple bounded weight $W_n^+$ modules. Any such module is isomorphic to the simple quotient of a tensor module $F(P,M)=P\otimes M$…

For every integer $k$ there exists a bound $B=B(k)$ such that if the characteristic polynomial of $g\in \operatorname{SL}_n(q)$ is the product of $\le k$ pairwise distinct monic irreducible polynomials over $\mathbb{F}_q$, then every…

I construct a correspondence between the Schubert cycles on the variety of complete flags in C^n and some faces of the Gelfand-Zetlin polytope associated with the irreducible representation of SL_n(C) with a strictly dominant highest…

We make progress on a question of Skandera by showing that a product of Kazhdan-Lusztig basis elements indexed by maximal elements of parabolic subgroups admits a Kazhdan-Lusztig basis element as a quotient arising from operations in the…

We study structural properties of truncated Weyl modules. A truncated Weyl module $W_N(\lambda)$ is a local Weyl module for $\mathfrak g[t]_N = \mathfrak g \otimes \frac{\mathbb C[t]}{t^N\mathbb C[t]}$, where $\mathfrak g$ is a…

Motivated by Lusztig's $G$-stable pieces, we consider the combinatorial pieces: the pairs $(w, K)$ for elements $w$ in the Weyl group and subsets $K$ of simple reflections that are normalized by $w$. We generalize the notion of cyclic shift…

We consider $d$-dimensional lattice polytopes $\Delta$ with $h^*$-polynomial $h^*_\Delta=1+h_k^*t^k$ for $1<k<(d+1)/2$ and relate them to some abelian subgroups of $\SL_{d+1}(\C)$ of order $1+h_k^*=p^r$ where $p$ is a prime number. These…