On Sevostyanov’s Construction of Quantum Difference Toda Lattices

We propose a natural generalization of the construction of the quantum difference Toda lattice [6, 22] associated with a simple Lie algebra $\mathfrak{g}$. Our construction depends on two orientations of the Dynkin diagram of $\mathfrak{g}$ and some other data (which we refer to as a pair of Sevostyanov triples). In types $A$ and $C$, we provide an alternative construction via Lax matrix formalism, cf. [15]. We also show that the generating function of the pairing of Whittaker vectors in the Verma modules is an eigenfunction of the corresponding modified quantum difference Toda system and derive fermionic formulas for the former in spirit of [7]. We give a geometric interpretation of all Whittaker vectors in type $A$ via line bundles on the Laumon moduli spaces and obtain an edge-weight path model for them, generalizing the construction of [4].

Recursions and divisibility properties for combinatorial Macdonald polynomials

Combinatorics International audience For each integer partition mu, let e (F) over tilde (mu)(q; t) be the coefficient of x(1) ... x(n) in the modified Macdonald polynomial (H) over tilde (mu). The polynomial (F) over tilde (mu)(q; t) can be regarded as the Hilbert series of a certain doubly-graded S(n)-module M(mu), or as a q, t-analogue of n! based on permutation statistics inv(mu) and maj(mu) that generalize the classical inversion and major index statistics. This paper uses the combinatorial definition of (F) over tilde (mu) to prove some recursions characterizing these polynomials, and other related ones, when mu is a two-column shape. Our result provides a complement to recent work of Garsia and Haglund, who proved a different recursion for two-column shapes by representation-theoretical methods. For all mu, we show that e (F) over tilde (mu)(q, t) is divisible by certain q-factorials and t-factorials depending on mu. We use our recursion and related tools to explain some of these factors bijectively. Finally, we present fermionic formulas that express e (F) over tilde ((2n)) (q, t) as a sum of q, t-analogues of n!2(n) indexed by perfect matchings.

TWO CHARACTER FORMULAS FOR $\widehat{\mathfrak{sl}}_2$ SPACES OF COINVARIANTS

We consider [Formula: see text] spaces of coinvariants with respect to two kinds of ideals of the enveloping algebra [Formula: see text]. The first one is generated by [Formula: see text], and the second one is generated by [Formula: see text], [Formula: see text] where P(t), [Formula: see text] are fixed generic polynomials. (We also treat a generalization of the latter.) Using a method developed in our previous paper, we give new fermionic formulas for their Hilbert polynomials in terms of the level-restricted Kostka polynomials and q-multinomial symbols. As a byproduct, we obtain a fermionic formula for the fusion product of [Formula: see text]-modules with rectangular highest weights, generalizing a known result for symmetric (or anti-symmetric) tensors.