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Constructions of representations of \(\text{o}(2n+1,{\mathbb C})\) that imply Molev and Reiner-Stanton lattices are strongly Sperner. (English) Zbl 1051.17009
Summary: Two infinite families of distributive lattices parameterized by positive integers n and \(k\) are considered. The first family of lattices, here denoted \(L_B^{\text{RS}}(k,2n)\), was introduced by V. Reiner and D. Stanton [J. Algebr. Comb. 7, No. 1, 91–107 (1998; Zbl 0935.05089)] as the distributive lattices Good\((k,2n)\) of certain partitions. There, Reiner and Stanton showed that these lattices are rank symmetric and rank unimodal and conjectured that they are strongly Sperner. The second family of lattices introduced here is denoted \(L_B^{\text{Mol}}(k,2n)\) because of its connection to certain representation constructions of the odd orthogonal Lie algebras obtained by A. I. Molev [J. Phys. A, Math. Gen. 33, No. 22, 4143–4158 (2000; Zbl 0988.17005)]. For fixed \(n\) and \(k\), the two lattices have the same rank generating function, but the lattices are isomorphic as posets if and only if \(k=1\).
In this paper, the lattices \(L_B^{\text{RS}}(k,2n)\) and \(L_B^{\text{Mol}}(k,2n)\) are used to produce two different constructions of the irreducible representation of the odd orthogonal Lie algebra \(o(2n+1, \mathbb C)\) isomorphic to the largest irreducible component in the \(k\)th symmetric power of the defining representation of \(o(2n+1, \mathbb C)\). Constructions of the analogous infinite family of irreducible representations of \(G_2\) are obtained as a special case. These constructions use the elements of the lattices to index bases for the representing spaces, and explicit formulas for the matrix entries of the representing matrices for certain Lie algebra generators are given. These constructions together with a result of Proctor imply that both lattices are rank symmetric, rank unimodal, and strongly Sperner.

MSC:
17B20 Simple, semisimple, reductive (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
06A07 Combinatorics of partially ordered sets
05E25 Group actions on posets, etc. (MSC2000)
20G05 Representation theory for linear algebraic groups
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