Related papers: Character Formulae of $\hat{sl}_n$-Modules and Inh…

We derive Rogers--Ramanujan type partition identities at the fundamental weight $\Lambda_0$ for the exceptional affine types $G_2^{(1)}$, $D_4^{(3)}$, $F_4^{(1)}$, $E_6^{(2)}$, $E_6^{(1)}$, $E_7^{(1)}$ and $E_8^{(1)}$. Our starting point is…

A $Q$-manifold $M$ is a supermanifold endowed with an odd vector field $Q$ squaring to zero. The Lie derivative $L_Q$ along $Q$ makes the algebra of smooth tensor fields on $M$ into a differential algebra. In this paper, we define and study…

Representations of small quantum groups $u_q({\mathfrak{g}})$ at a root of unity and their extensions provide interesting tensor categories, that appear in different areas of algebra and mathematical physics. There is an ansatz by Lusztig…

We study evaluation modules for quantum symmetric pair coideal subalgebras of affine type $\mathsf{AI}$. By computing the action of the generators in Lu and Wang's Drinfeld-type presentation on Gelfand-Tsetlin bases, we determine the…

We fix integers $k> 0$ and $n>0$. For a $k$-punctured Riemann surface $\Sigma \setminus \{ p_1,\ldots,p_k \}$ and a $k$-tuple $\boldsymbol{\mu}=(\mu^1,\ldots,\mu^k)$ of partitions of $n$, we can define the character variety of type…

Let $S_n$ and $S_{n,k}$ be, respectively, the number of subsets and $k$-subsets of $\mathbb{N}_n=\{1,\ldots,n\}$ such that no two subset elements differ by an element of the set $\mathcal{Q}$, the largest element of which is $q$. We prove a…

Fermionic formulae originate in the Bethe ansatz in solvable lattice models. They are specific expressions of some q-polynomials as sums of products of q-binomial coefficients. We consider the fermionic formulae associated with general…

We introduce the notion of characters of comodules over coribbon Hopf algebras. The case of quantum groups of type $A_n$ is studied. We establish a characteristic equation for the quantum matrix and a q-analogue of Harish-Chandra-…

We define three combinatorial models for \hat{sl(n)} crystals, parametrized by partitions, configurations of beads on an `abacus', and cylindric plane partitions, respectively. These are reducible, but we can identify an irreducible…

Let $\pi_1,...,\pi_n$ be an irreducible finite-dimensional $\mathfrak{sl}_2$-modules. Using the theory of the representations of the current algebras, we introduce a several ways to construct a $q$-grading on $\pi_1\otimes...\otimes\pi_n$.…

Using face algebras (i.e. algebras of L-operators of IRF models), we construct modular tensor categories with positive definite inner product, whose fusion rules and S-matrices are the same as (or slightly different from) those obtained by…

For each integer $n\geq 1$, we construct a $\operatorname{GL}_n(\mathbb Q)$-invariant modular symbol $\bm\xi_n$ with coefficients in a space of distributions that takes values in the Milnor $K_n$-group of the modular function field. The…

Let B be a commutative B\'ezout domain B and let MSpec(B) be the maximal spectrum of B. We obtain a Feferman-Vaught type theorem for the class of B-modules. We analyse the definable sets in terms, on one hand, of the definable sets in the…

In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety $\Zl$ in a quiver variety, and show the following results: (1) The…

Let F* be the field of q elements and let P(n,q) denote the projective space of dimension n-1 over F*. We construct a family H^{n}_{k,i} of combinatorial homology modules associated to P(n,q) for a coefficient field F of positive…

A representation of the quantum affine algebra $U_{q}(\hat{sl_{2}})$ of an arbitrary level $k$ is realized in terms of three boson fields, whose $q \rightarrow 1$ limit becomes the Wakimoto representation. An analogue of the screening…

It is shown that path algebras modulo relations of the form $\Lambda = KQ/I$, where $Q$ is a quiver, $K$ a coefficient field, and $I \subseteq KQ$ the ideal generated by all paths of a given length, can be readily analyzed homologically,…

Let $B(\Lambda)$ be a level $\ell$ highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\gB(\Lambda)$ of the crystal $B(\Lambda)$ using quiver…

Let ${\bf G}$ be a connected reductive group over $\bar{\mathbb{F}}_q$, the algebraically closure of $\mathbb{F}_q$ (the finite field with $q=p^e$ elements), with the standard Frobenius map $F$. Let ${\bf B}$ be an $F$-stable Borel…

We show that there exists a suitable sequence $\{w^{(k)}\}_{k \ge 0}$ of Weyl group elements for the perfect crystal $B = B^{1,3l}$ such that the path realizations of the Demazure crystals $B_{w^{(k)}}(l\Lambda_2)$ for the quantum affine…