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 $⟨\mu ,{\alpha }_2^{\vee }⟩=1$ and $\mu +\beta$ dominant, $a(\mu ,\mu +\beta )$ is $-2$ if $⟨\mu ,{\alpha }_1^{\vee }⟩=0$ 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.

47 pages, 2 figures