Atomic Decompositions of Lie Characters and the Dominant Weight Poset

Abstract

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and, for a dominant integral weight $\lambda$, let $\Theta_\lambda$ be the multiplicity-free sum of the weights of the irreducible module $V(\lambda)$. The $\Theta_\mu$ form a $\mathbb{Z}$-basis of the $W$-invariants, so there are unique integers $a(\mu,\lambda)$, the atomic numbers, with $\mathrm{ch}\,V(\lambda) = \sum_\mu a(\mu,\lambda)\Theta_\mu$ over dominant $\mu$. We ask when they are nonnegative. They are the Moebius transform of the weight-multiplicity function on the dominant-weight poset, which the crosscut theorem turns into an alternating sum of at most $2^{\mathrm{rank}\,\Phi}$ multiplicities indexed by subsets of the covers of $\mu$ in $[\mu,\lambda]$. We prove that $a(\mu,\lambda)\ge 0$ whenever every connected component of the Dynkin diagram is a path, and identify the coefficient with the dimension of an explicit weight space: it is cut out of $V(\lambda)$ by alternately taking kernels of raising operators and cokernels of lowering operators, one per cover, in order along the path. Type $D_4$ shows the hypothesis is necessary. Put $\beta = \alpha_1 + 2\alpha_2 + 2\alpha_3 + 2\alpha_4$; for every dominant $\mu$ with $\langle\mu,\alpha_2^\vee\rangle = 1$ and $\mu+\beta$ dominant, $a(\mu,\mu+\beta)$ is $-2$ if $\langle\mu,\alpha_1^\vee\rangle = 0$ and $-1$ otherwise. Restriction to the support of $\lambda-\mu$ carries this family into every irreducible type with a trivalent node. Thus an irreducible finite root system has all atomic numbers nonnegative exactly when its Dynkin diagram is a path, namely in types $A_n$, $B_n$, $C_n$, $F_4$, $G_2$. Deep in the dominant chamber, in an explicit range, $a(\mu,\lambda)$ is the Kostant partition number of $\lambda-\mu$ for the nonsimple positive roots; negative atomic numbers are therefore confined to boundary slabs.