A combinatorial formula for graded multiplicities in excellent filtrations

A filtration of a representation whose successive quotients are isomorphic to Demazure modules is called an excellent filtration. In this paper we study graded multiplicities in excellent filtrations of fusion products for the current algebra $\mathfrak{sl}_2[t]$. We give a combinatorial formula for the polynomials encoding these multiplicities in terms of two dimensional lattice paths. Corollaries to our main theorem include a combinatorial interpretation of various objects such as the coeffficients of Ramanujan's fifth order mock theta functions $\phi_0, \phi_1, \psi_0, \psi_1$, Kostka polynomials for hook partitions and quotients of Chebyshev polynomials. We also get a combinatorial interpretation of the graded multiplicities in a level one flag of a local Weyl module associated to the simple Lie algebras of type $B_n \text{ and } G_2$

Demazure Filtrations of Tensor Product Modules and Character Formula

We study the structure of the finite-dimensional representations of $\mathfrak{sl}_2[t]$, the current Lie algebra type of $A_1$, which are obtained by taking tensor products of special Demazure modules. We show that these representations admit a Demazure flag and obtain a closed formula for the graded multiplicities of the level 2 Demazure modules in the filtration of the tensor product of two local Weyl modules for $\mathfrak{sl}_2[t]$. Furthermore, we derive an explicit expression for graded character of the tensor product of a local Weyl module with an irreducible $\mathfrak{sl}_2[t]$ module. In conjunction with the results of \cite{MR3210603}, our findings provide evidence for the conjecture in \cite{9} that the tensor product of Demazure modules of levels m and n respectively has a filtration by Demazure modules of level m + n

Macdonald Polynomials and level two Demazure modules for affine sln+1\mathfrak{sl}_{n+1}

We define a family of symmetric polynomials $G_{\nu,\lambda}(z_1,\cdots, z_{n+1},q)$ indexed by a pair of dominant integral weights. The polynomial $G_{\nu,0}(z,q)$ is the specialized Macdonald polynomial and we prove that $G_{0,\lambda}(z,q)$ is the graded character of a level two Demazure module associated to the affine Lie algebra $\widehat{\mathfrak{sl}}_{n+1}$. Under suitable conditions on $(\nu,\lambda)$ (which includes the case when $\nu=0$ or $\lambda=0$) we prove that $G_{\nu,\lambda}(z,q)$ is Schur positive and give explicit formulae for them in terms of Macdonald polynomials

Macdonald Polynomials and Level Two Demazure Modules for Affine sln+1

We define a family of symmetric polynomials Gν,λ(z1, ...,zn+1, q) indexed by a pair of dominant integral weights for a root system of type An. The polynomial Gν,0(z, q) is the specialized Macdonald polynomial Pν(z, q, 0) and is known to be the graded character of a level one Demazure module associated to the affine Lie algebra sln+1. We prove that G0,λ(z, q) is the graded character of a level two Demazure module for sln+1. Under suitable conditions on (ν, λ) (which apply to the pairs (ν, 0) and (0, λ)) we prove that Gν,λ(z, q) is Schur positive, i.e., it can be written as a linear combination of Schur polynomials with coefficients in Z+[q]. We further prove that Pν(z, q, 0) is a linear combination of elements G0,λ(z, q) with the coefficients being essentially products of q-binomials. Together with a result of K. Naoi, a consequence of our result is an explicit formula for the specialized Macdonald polynomial associated to a non-simply laced Lie algebra as a linear combination of the level one Demazure characters of the non-simply laced algebra

Módulos de Weyl truncados via módulos de Chari-Venkatesh e produtos de fusão

Orientador: Moura, Adriano Adrega deTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Estudamos propriedades estruturais de módulos de Weyl truncados. Dados uma álgebra de Lie simples g e um peso integral dominante \lambda , o módulo de Weyl local graduado W(\lambda ) é o objeto universal na categoria dos módulos de dimensão finita graduados de peso máximo para a ágebra de correntes g[t]= g \otimes C[t]. Para cada inteiro positivo N, o quociente W_N(\lambda ) de W(\lambda ) pelo submódulo gerado pela ação do ideal g \otimes t^NC[t] sobre o vetor de peso máximo é chamado um módulo de Weyl truncado. Ele satisfaz a mesma propriedade universal de W(\lambda ) quando visto como um módulo para a correspondente álgebra de correntes truncada g[t]_N= g \otimes \frac{ C[t]}{t^NC[t]}. Chari-Fourier-Sagaki conjecturaram que se N \leq |\lambda |, W_N(\lambda) deve ser isomorfo a um produto de fusão de certos módulos irredutíveis. Nosso principal resultado prova essa conjectura quando \lambda é um múltiplo de um peso minúsculo e g é de tipo ADE. Também damos um passo adiante para provar a conjectura para múltiplos de um peso fundamental "pequeno" que não é minúsculo provando que o módulo de Weyl truncado correspondente é isomorfo ao quociente de um produto de fusão de módulos de Kirillov-Reshetikhin por uma simples relação. Uma parte importante da demonstração de nosso resultado principal é dedicada a provar que qualquer módulo de Weyl truncado é isomorfo a um módulo de Chari-Venkatesh com a correspondente família de partições explicitamente descrita. Este fato é o segundo resultado principal deste trabalho e nos leva a novos resultados no caso g = {sl}_2 relacionados a bandeiras de Demazure e cadeias de inclusões de Módulos de Weyl truncadosAbstract: We study structural properties of truncated Weyl modules. Given a simple Lie algebra g and a dominant integral weight \lambda, the graded local Weyl module W(\lambda) is the universal finite-dimensional graded highest-weight module for the current algebra g[t]= g\otimes C[t]. For each positive integer N, the quotient W_N(\lambda) of W(\lambda) by the submodule generated by the action of the ideal g\otimes t^NC[t] on the highest-weight vector is called a truncated Weyl module. It satisfies the same universal property as W(\lambda) when regarded as a module for the corresponding truncated current algebra g[t]_N=g \otimes \frac{C[t]}{t^NC[t]}. Chari-Fourier-Sagaki conjectured that if N\leq |\lambda|, W_N(\lambda) should be isomorphic to the fusion product of certain irreducible modules. Our main result proves this conjecture when \lambda is a multiple of a minuscule weight and g is simply laced. We also take a further step towards proving the conjecture for multiples of a ''small'' fundamental weight which is not minuscule by proving that the corresponding truncated Weyl module is isomorphic to the quotient of a fusion product of Kirillov-Reshetikhin modules by a very simple relation. One important part of the proof of our main result, and the second main result of this work, is a proof that any truncated Weyl module is isomorphic to a Chari-Venkatesh module and explicitly describes the corresponding family of partitions. This leads to further results in the case that g={sl}_2 related to Demazure flags and chains of inclusions of truncated Weyl modulesDoutoradoMatematicaDoutor em Matemática140768/2016-5CNPQCAPE