Permutation Weights for Affine Lie Algebras

In most applications of semi-simple Lie groups and algebras representation theory, calculating weight multiplicities is one of the most often used and effort consuming operations. The existing tools were created many years ago by Kostant and Freudenthal. The celebrated Kostant weight multiplicity formula uses summation over the Weyl group of values of Kostant partition function, and the Freudenthal formula is recurrent. In this paper, a new way for calculating weight multipli...

The recursion relations of branching coefficients $k_{\xi}^{(\mu)}$ for a module $L_{\frak{g}\downarrow \frak{h}}^{\mu}$ reduced to a Cartan subalgebra $\frak{h}$ are transformed in order to place the recursion shifts $\gamma \in \Gamma_{\frak{a}\subset \frak{h}}$ into the fundamental Weyl chamber. The new ensembles $F\Psi$ (the "folded fans") of shifts were constructed and the corresponding recursion properties for the weights belonging to the fundamental Weyl chamber were f...

The center of an extended affine Hecke algebra is known to be isomorphic to the ring of symmetric functions associated to the underlying finite Weyl group $W\_0$. The set of Weyl characters ${\sf s}\_\la$ forms a basis of the center and Lusztig showed in [Lus15] that these characters act as translations on the Kazhdan-Lusztig basis element $C\_{w\_0}$ where $w\_0$ is the longest element of $W\_0$, that is we have $C\_{w\_0}{\sf s}\_\la =C\_{w\_0t\_\la}$. As a consequence, the...

In this paper, we classify all irreducible weight modules with finite-dimensional weight spaces over the affine-Virasoro Lie algebra of type $A_1$.

We study generating functions for Lusztig's $t$-analog of weight multiplicities associated to integrable highest weight representations of the simplest affine Lie algebra $A_1^{(1)}$. At $t=1$, these reduce to the {\em string functions} of $A_1^{(1)}$, which were shown by Kac and Peterson to be related to certain Hecke indefinite modular forms. Using their methods, we obtain a description of the general $t$-string function; we show that its values can be realized as radial av...

Let $\Lambda$ be a dominant integral weight of level $k$ for the affine Lie algebra $\mathfrak g$ and let $\alpha$ be a non-negative integral combination of simple roots. We address the question of whether the weight $\eta=\Lambda-\alpha$ lies in the set $P(\Lambda)$ of weights in the irreducible highest-weight module with highest weight $\Lambda$. We give a non-recursive criterion in terms of the coefficients of $\alpha$ modulo an integral lattice $kM$, where $M$ is the latt...

It is shown that there are infinitely many formulas to calculate multiplicities of weights participating in irreducible representations of $A_N$ Lie algebras. On contrary to recursive character of Kostant and Freudenthal multiplicity formulas, they provide us systems of linear algebraic equations with N-dependent polinomial coefficients. These polinomial coefficients are in fact related with polinomials which represent eigenvalues of Casimir operators.

The multiplicity of a weight $\mu$ in an irreducible representation of a simple Lie algebra $\mathfrak{g}$ with highest weight $\lambda$ can be computed via the use of Kostant's weight multiplicity formula. This formula is an alternating sum over the Weyl group and involves the computation of a partition function. In this paper we consider a $q$-analog of Kostant's weight multiplicity and present a SageMath program to compute $q$-multiplicities for the simple Lie algebras.

We expand the affine Weyl denominator formulas as signed $q$-series of ordinary Weyl characters running over the affine Grassmannian. Here the grading in $q$ coincides with the (dual) atomic length of the root system considered as introduced by Chapelier-Laget and Gerber. Next, we give simple expressions of the atomic lengths in terms of self-conjugate core partitions. This permits in particular to rederive, from the general theory of affine root systems, some results of the ...

We prove that the multiplicities of certain maximal weights of $\mathfrak{g}(A^{(1)}_{n})$-modules are counted by pattern avoidance on words. This proves and generalizes a conjecture of Misra-Rebecca. We also prove similar phenomena in types $A^{(2)}_{2n}$ and $D^{(2)}_{n+1}$. Both proofs are applications of Kashiwara's crystal theory.