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# Title: Layer structure of irreducible Lie algebra modules

Abstract: Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra. A layer sum is introduced as the sum of formal exponentials of the distinct weights appearing in an irreducible $\mathfrak{g}$-module. It is argued that the character of every finite-dimensional irreducible $\mathfrak{g}$-module admits a decomposition in terms of layer sums, with only non-negative integer coefficients. Ensuing results include a new approach to the computation of Weyl characters and weight multiplicities, and a closed-form expression for the number of distinct weights in a finite-dimensional irreducible $\mathfrak{g}$-module. The latter is given by a polynomial in the Dynkin labels, of degree equal to the rank of $\mathfrak{g}$.
 Comments: 23 pages Subjects: Representation Theory (math.RT); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Combinatorics (math.CO) Cite as: arXiv:1803.06592 [math.RT] (or arXiv:1803.06592v1 [math.RT] for this version)

## Submission history

From: Jorgen Rasmussen [view email]
[v1] Sun, 18 Mar 2018 02:40:00 GMT (17kb)