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We present a conjecture generalizing the Cauchy formula for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials of the character varieties of representations of the fundamental group of a Riemann surface of genus g to…
We prove the Kirillov-Reshetikhin conjecture concerning certain finite dimensional representations of a quantum affine algebra $\Ua$ when $\hat\g$ is an untwisted affine Lie algebra of type $ADE$. We use $t$--analog of $q$--characters…
Generalized Jacobi polynomials are orthogonal polynomials related to a weight function which is smooth and positive on the whole interval of orthogonality up to a finite number of points, where algebraic singularities occur. The influence…
A notion of Drinfeld polynomials is introduced for modules of two-parameter quantum affine algebras. Finite dimensional representations are then characterized by sets of $l$-tuples of pairs of Drinfeld polynomials with certain conditions.
In this paper we classify and give Kazhdan-Lusztig type character formulas for equivariantly irreducible representations of Lie algebras of reductive algebraic groups over a field of large positive characteristic. The equivariance is with…
We produce 2-representations of the positive part of affine quantum enveloping algebras on their finite-dimensional counterparts in type $A_n$. These 2-representations naturally extend the right-multiplication 2-representation of…
We consider algebras over a field K defined by a presentation K <x_1,..., x_n : R >, where $R$ consists of n choose 2 square-free relations of the form x_i x_j = x_k x_l with every monomial x_i x_j, i different from j, appearing in one of…
We consider various specializations of the non-twisted quantum affine algebras at roots of unity. We define and study the q-characters of their finite-dimensional representations.
We study connections between the ring of symmetric functions and the characters of irreducible finite-dimensional representations of quantum affine algebras. We study two families of representations of the symplectic and orthogonal Lie…
We define the cluster algebra associated with the Q-system for the Kirillov-Reshetikhin characters of the quantum affine algebra $U_q(\hat{\g})$ for any simple Lie algebra g, generalizing the simply-laced case treated in [Kedem 2007]. We…
We develop the theory of $q$-characters for quantum affine superalgebras of type $A$ in connection with deformed Cartan matrices. To achieve this, we establish a Khoroshkin-Tolstoy-type multiplicative formula of the universal $R$-matrix of…
The q-characters were introduced by Frenkel and mReshetikhin to study finite dimensional representations of the untwisted quantum affine algebra for q generic. The $\epsilon$-characters at roots of unity were constructed by Frenkel and…
We derive some q-analogs of Euler-Cassini-type identities and of recurrence formulas for powers of Fibonacci polynomials.
In this note, we give a new proof of the fact that an affine semiprime algebra R of Gelfand-Kirillov dimension 1 satisfies a polynomial identity. Our proof uses only the growth properties of the algebra and yields an explicit upper bound…
Jacobi-Trudy formula for a generalisation of Schur polynomials related to any sequence of orthogonal polynomials in one variable is given. As a corollary we have Giambelli formula for generalised Schur polynomials.
Goulden-Rattan polynomials give the exact value of the subdominant part of the normalized characters of the symmetric groups in terms of certain quantities ($C_i$) which describe the macroscopic shape of the Young diagram. The…
We consider the following question, concerning associative algebras R over an algebraically closed field k: When can the space of (equivalence classes of) finite dimensional irreducible representations of R be topologically embedded into a…
We use $q$-binomial theorem to prove three new polynomial identities involving $q$-trinomial coefficients. We then use summation formulas for the $q$-trinomial coefficients to convert our identities into another set of three polynomial…
We formulate several polynomial identities. One side of these identities has a nice simple form. Whereas the other has a form of a polynomial whose coefficients contain binomial coefficients double factorials or (and) rising factorials. The…
This paper deals with the variety of commutative algebras satisfying one of the four families of degree four identities found by Carini, Hentzel and Piacentini-Cattaneo in 1988. We give a characterization of representations and irreducible…