Robert G. Donnelly

Extremal Properties of Bases for Representations of Semisimple Lie Algebras

Let ℒ be a complex semisimple Lie algebra with specified Chevalley generators. Let V be a finite dimensional representation of ℒ with weight basis ℬ. The supporting graph P of ℬ is defined to be the directed graph whose vertices are the elements of ℬ and whose colored edges describe the supports of the actions of the Chevalley generators on V. Four properties of weight bases are introduced in this setting, and several families of representations are shown to have weight bases which have or are conjectured to have each of the four properties. The basis ℬ can be determined to be edge-minimizing (respectively, edge-minimal) by comparing P to the supporting graphs of other weight bases of V. The basis ℬ is solitary if it is the only basis (up to scalar changes) which has P as its supporting graph. The basis ℬ is a modular lattice basis if P is the Hasse diagram of a modular lattice. The Gelfand-Tsetlin bases for the irreducible representations of sl(n, ℂ) serve as the prototypes for the weight bases sought in this paper. These bases, as well as weight bases for the fundamental representations of sp(2n, ℂ) and the irreducible “one-dimensional weight space” representations of any semisimple Lie algebra, are shown to be solitary and edge-minimal and to have modular lattice supports. Tools developed here are used to construct uniformly the irreducible one-dimensional weight space representations. Similar results for certain irreducible representations of the odd orthogonal Lie algebra o(2n + 1, ℂ), the exceptional Lie algebra G2, and for the adjoint and short adjoint representations of the simple Lie algebras are announced.

Constructions of representations of o(2 n + 1, C) that imply Molev and Reiner-Stanton lattices are strongly Sperner

Two infinite families of distributive lattices parameterized by positive integers n and k are considered. The first family of lattices, here denoted LRSB(k,2n), was introduced by Reiner and Stanton (J. Algebraic Combin. 7 (1998) 91) 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 LMolB(k, 2n) because of its connection to certain representation constructions of the odd orthogonal Lie algebras obtained by Molev (J. Phys. A 33 (2000) 4143). 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 LRSB(k,2n) and LMolB(k,2n) are used to produce two different constructions of the irreducible representation of the odd orthogonal Lie algebra o(2n + 1, C) isomorphic to the largest irreducible component in the kth symmetric power of the defining representation of o(2n + 1, C). Constructions of the analogous infinite family of irreducible representations of G2 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.

Distributive Lattices Defined for Representations of Rank Two Semisimple Lie Algebras

For a rank two root system and a pair of nonnegative integers, using only elementary combinatorics we construct two posets. The constructions are uniform across the root systems

Symplectic Analogs of the Distributive Lattices L(m, n)

A_{1}\oplus A_{1}

Extremal Bases for the Adjoint Representations of the Simple Lie Algebras

,

Root Systems for Asymmetric Geometric Representations of Coxeter Groups

A_{2}

Powerball, Expected Value, and the Law of (Very) Large Numbers.

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Explicit Constructions of the Fundamental Representations of the Symplectic Lie Algebras

C_{2}

The numbers game and Dynkin diagram classication results

, and

Eriksson's numbers game and finite Coxeter groups

G_{2}