相关论文: A combinatorial proof of a Weyl type formula for h…
A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. Ram and…
A theorem due to Tokuyama expresses Schur polynomials in terms of Gelfand-Tsetlin patterns, providing a deformation of the Weyl character formula and two other classical results, Stanley's formula for the Schur $q$-polynomials and Gelfand's…
We prove a formula for $S_n$ characters which are indexed by the partitions in the $(k,\ell)$ hook. The proof applies a combinatorial part of the theory of Lie superalgebras.
We examine and present new combinatorics for the Schur polynomials from the viewpoint of quantum integrability. We introduce and analyze an integrable six-vertex model which can be viewed as a certain degeneration model from a t-deformed…
We calculate the cohomology of $\mathfrak{sl}_3(k)$ over an algebraically closed field $k$ of characteristic $p>3$ with coefficients in simple modules and Weyl modules. We also give descriptions of the corresponding cohomology of…
In this paper the general Krall-type Askey-Wilson polynomials are introduced. These polynomials are obtained from the Askey-Wilson polynomials via the addition of two mass points to the weight function of them at the points $\pm1$. Several…
Let A a k-algebra, H a Hopf algebra, E = A#H a general crossed product and M an E-bimodule. We obtain a complex simpler than the canonical one, giving the Hochschild homology of E with coefficients in M. This complex is eqquiped with a…
Over a field $F$ of any characteristic, for a commutative associative algebra $A$ with an identity element and for the polynomial algebra $F[D]$ of a commutative derivation subalgebra $D$ of $A$, the associative and the Lie algebras of Weyl…
Higher order cohomology of arithmetic groups is expressed in terms of (g,K)-cohomology. Generalizing results of Borel, it is shown that the latter can be computed using functions of (uniform) moderate growth. A higher order versions of…
We present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. When a certain commutative subalgebra is finitely generated over an algebraically closed field we obtain a classification…
We introduce the new combinatorial approach of plethystic type of tableaux, as a method to understand coefficients of Schur functions appearing in plethysms $s_\nu[h_\lambda]$ and $s_{\nu}[e_{\lambda}]$, for any partitions $\lambda$ and…
We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases $t=1$ and $q=0$,…
Let G be a complex, connected, reductive algebraic group with Weyl group W and Steinberg variety Z. We show that the graded Borel-Moore homology of Z is isomorphic to the smash product of the coinvariant algebra of W and the group algebra…
We compute an explicit formula the Hilbert (Poincar\'e) series for the ring of hook Schur functions and (equivalently) the generating function for partitions which fit in a $(k,l)$-hook.
We consider two examples of a fully decodable combinatorial encoding of Bernoulli schemes: the encoding via Weyl simplices and the much more complicated encoding via the RSK (Robinson--Schensted--Knuth) correspondence. In the first case,…
We establish combinatorial and inductive formulas for Kazhdan-Lusztig polynomials associated to covexillary elements in classical types, extending results of Boe, Lascoux-Sch\"{u}tzenberger, Sankaran-Vanchinathan, and Zelevinsky for…
We show that the higher-order Weyl algebras over a field of characteristic zero, which are formally rigid as associative algebras, can be formally deformed in a nontrivial way as hom-associative algebras. We also show that these…
We compute the cohomology of the complement of toric arrangements associated to root systems as representations of the corresponding Weyl groups. Specifically, we develop an algorithm for computing the cohomology of the complement of toric…
We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials,…
In this paper, we study the Littlewood theory associated with the quantum super immanants and supersymmetric polynomials, including both the super case and the quantum generalization. In the setting of quantum super Schur-Weyl duality…