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This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy. Starting from second order difference equations we move on to self-adjoint operators and develop… (More)

- Georgia Benkart, Frank Sottile, Jeffrey Stroomer
- J. Comb. Theory, Ser. A
- 1996

We deene and characterize switching, an operation that takes two tableaux sharing a common border and \moves them through each other" giving another such pair. Several authors, including James and Kerber, Remmel, Haiman, and Shimozono, have deened switching operations; however, each of their operations is somewhat diierent from the rest and each imposes a… (More)

- Georgia Benkart, Tom Roby
- 1998

The algebra generated by the down and up operators on a differential partially ordered set (poset) encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on posets, we introduce here a family of infinite-dimensional associative algebras called down-up algebras. We show that… (More)

- Georgia Benkart, Sarah Witherspoon, GEORGIA BENKART, SARAH WITHERSPOON
- 2004

We determine the finite-dimensional simple modules for two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl n and sln, and give a complete reducibility result. These quantum groups have a natural n-dimensional module V . We prove an analogue of Schur-Weyl duality in this setting: the centralizer algebra of the… (More)

- GEORGIA BENKART, DUNCAN MELVILLE
- 1998

We construct quantum deformations of enveloping algebras of Borcherds superalgebras, their Verma modules, and their irreducible highest weight modules.

Two classes of avalanche-finite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chip-firing/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKay-Cartan matrices for finite subgroups G of general linear groups. In the root system case, the recurrent and superstable configurations are… (More)

- Georgia Benkart, Sarah Witherspoon, GEORGIA BENKART, SARAH WITHERSPOON
- 2008

We investigate two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gln and sln. We show that these quantum groups can be realized as Drinfel’d doubles of certain Hopf subalgebras with respect to Hopf pairings. Using the Hopf pairing, we construct a corresponding R-matrix. The quantum groups have a natural… (More)

We develop general results on centroids of Lie algebras and apply them to determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody Lie algebras, and Lie algebras graded by finite root systems.

- Georgia Benkart
- 2008

We consider the three-point loop algebra, L = sl2 ⊗ F[t, t −1 , (t − 1)−1], where F denotes a field of characteristic 0 and t is an indeterminate. The universal central extension L̂ of L was determined by Bremner. In this note, we give a presentation for L̂ via generators and relations, which highlights a certain symmetry over the alternating group A4. To… (More)