Highest Weight Vectors in Plethysms

Tau-functions as highest weight vectors for W1+∞ algebra

For each r = (r1, r2, . . . , rN ) ∈ C N we construct a highest weight module Mr of the Lie algebra W1+∞. The highest weight vectors are specific tau-functions of the N -th Gelfand–Dickey hierarchy. We show that these modules are quasifinite and we give a complete description of the reducible ones together with a formula for the singular vectors. hep-th/9510211

Even Partitions in Plethysms

We prove that for all natural numbers k, n, d with k ≤ d and every partition λ of size kn with at most k parts there exists an irreducible GLd(C)-representation of highest weight 2λ in the plethysm Sym(SymC). This gives an affirmative answer to a conjecture by Weintraub (J. Algebra, 129 (1): 103-114, 1990). Our investigation is motivated by questions of geometric complexity theory and uses idea...

Bethe states as highest weight vectors of the sl 2 loop algebra at roots of unity

We show that regular Bethe ansatz eigenvectors of the XXZ spin chain at roots of unityare highest weight vectors and generate irreducible representations of the sl2 loop algebra.Here the parameter q, which is related to the XXZ anisotropy ∆ through ∆ = (q+q−1)/2,is given by a root of unity, q2N = 1, for an integer N . First, for a regular Bethe stateat a root of unity, we sh...

HIGHEST-WEIGHT VECTORS FOR THE ADJOINT ACTION OF GLn ON POLYNOMIALS, II

HIGHEST WEIGHT VECTORS OF IRREDUCIBLE REPRESENTATIONS OF THE QUANTUM SUPERALGEBRA Uq (gl(m,n))

The Iwahori-Hecke algebra Hk(q ) of type A acts on tensor product space V ⊗k of the natural representation of the quantum superalgebra Uq(gl(m, n)). We show this action of Hk(q ) and the action of Uq(gl(m, n)) on the same space determine commuting actions of each other. Together with this result and Gyoja’s q-analogue of the Young symmetrizer, we construct a highest weight vector of each irredu...