Related papers: Combinatorial relations among relations for level …

We construct a family of exact functors from the BGG category of representations of the Lie algebra sl to the category of finite-dimensional representations of the degenerate (or graded) affine Hecke algebra H of GL. These functors…

We investigate the representation theory of domestic group schemes $\mathcal{G}$ over an algebraically closed field of characteristic $p > 2$. We present results about filtrations of induced modules, actions on support varieties, Clifford…

We introduce a class of induced representations of the degenerate double affine Hecke algebra of gl_N and analyze their structure mainly by means of intertwiners. We also construct them from modules of the affine Lie algebra using…

We work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local…

We study the category $\mathcal{F}_n$ of finite-dimensional integrable representations of the periplectic Lie superalgebra $\mathfrak{p}(n)$. We define an action of the Temperley--Lieb algebra with infinitely many generators and defining…

One can realize higher laminations as positive configurations of points in the affine building. The duality pairings of Fock and Goncharov give pairings between higher laminations for two Langlands dual groups $G$ and $G^{\vee}$. These…

Certain combinatorial bases of Feigin-Stoyanovsky's type subspaces of level k standard modules for affine Lie algebra sl(r,C)\sptilde are parametrized by (k,r)-admissible configurations. In this note we use Capparelli-Lepowsky-Milas' method…

In 1983, Feingold and Frenkel discovered a relation between Siegel modular forms of genus two and a rank-three hyperbolic Kac--Moody algebra extending the affine Lie algebra of type $A_1$. It inspires a problem to explore more general…

Given a category C of a combinatorial nature, we study the following fundamental question: how does the combinatorial behavior of C affect the algebraic behavior of representations of C? We prove two general results. The first gives a…

The problem of determining maximal ideals in universal affine vertex algebras is difficult for levels beyond admissible, since there are no simple character formulas which can be applied. Here we investigate when certain quotient $\mathcal…

We develop a general approach to finding combinatorial models for cluster algebras. The approach is to construct a labeled graph called a framework. When a framework is constructed with certain properties, the result is a model…

We completely determine the structure constants between real root vectors in a rank 2 Kac--Moody algebra $\mathfrak{g}$. Our description is computationally efficient, even in the rank 2 hyperbolic case where the coefficients of roots on the…

We classify contravariant pairings between standard Whittaker modules and Verma modules over a complex semisimple Lie algebra. These contravariant pairings are useful in extending several classical techniques for category $\mathcal{O}$ to…

This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [6], [7], [8]. Here we consider conformal embeddings in simple affine vertex superalgebra $V_k(\mathfrak g)$ where $\mathfrak g=\mathfrak…

Let $\mathfrak g$ be a semisimple Lie algebra and $\mathfrak k\subset\mathfrak g$ be a reductive in $\mathfrak g$ subalgebra. A $(\mathfrak g, \mathfrak k)$-module is a $\mathfrak g$-module which after restriction to $\mathfrak k$ becomes a…

In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our…

We show that, for many Lie superalgebras admitting a compatible $\mathbb{Z}$-grading, Kac induction functor gives rise to a bijection between simple supermodules over a Lie superalgebra and simple supermodules over the even part of this Lie…

We study the annihilator of the cokernel of a map of free Z/2-graded modules over a Z/2-graded skew-commutative algebra in characteristic 0 and define analogues of its Fitting ideals. We show that in the ``generic'' case the annihilator is…

We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n,C). Our main results are: 1) Pieri rules for the Schubert bases of H^*(Gr) and H_*(Gr), which expresses the product of a special…

We classify all uniserial modules of the solvable Lie algebra $\mathfrak{g}=\langle x\rangle \ltimes V$, where $V$ is an abelian Lie algebra over an algebraically closed field of characteristic 0 and $x$ is an arbitrary automorphism of $V$.