On a bijection between Littlewood-Richardson fillings of conjugate shape

We present a new bijective proof of the equality between the number of Littlewood-Richardson fillings of a skew-shape [lambda]/[mu] of weight [nu], and those of the conjugate skew-shape [lambda]t/[mu]t, of conjugate weight [nu]t. The bijection is defined by means of a unique permutation [alpha][lambda]/[mu] associated to the skew-shape [lambda]/[mu]. Our arguments use only well-established properties of Schensted insertion, and make no reference to jeu de taquin.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30058/1/0000426.pd

Identities for classical group characters of nearly rectangular shape

We derive several identities that feature irreducible characters of the general linear, the symplectic, the orthogonal, and the special orthogonal groups. All the identities feature characters that are indexed by shapes that are "nearly" rectangular, by which we mean that the shapes are rectangles except for one row or column that might be shorter than the others. As applications we prove new results in plane partitions and tableaux enumeration, including new refinements of the Bender-Knuth and MacMahon (ex-)conjectures.Comment: 55 pages, AmS-TeX; to appear in J. Algebr

Combinatorial representation theory of Lie algebras. Richard Stanley's work and the way it was continued

Richard Stanley played a crucial role, through his work and his students, in the development of the relatively new area known as combinatorial representation theory. In the early stages, he has the merit to have pointed out to combinatorialists the potential that representation theory has for applications of combinatorial methods. Throughout his distinguished career, he wrote significant articles which touch upon various combinatorial aspects related to representation theory (of Lie algebras, the symmetric group, etc.). I describe some of Richard's contributions involving Lie algebras, as well as recent developments inspired by them (including some open problems), which attest the lasting impact of his work.Comment: 11 page

Row-strict quasisymmetric Schur functions

Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions called the quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through reverse column-strict tableaux. We introduce a new basis for quasisymmetric functions called the row-strict quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through row-strict tableaux. We describe the relationship between this new basis and other known bases for quasisymmetric functions, as well as its relationship to Schur polynomials. We obtain a refinement of the omega transform operator as a result of these relationships.Comment: 17 pages, 11 figure

Skew row-strict quasisymmetric Schur functions

Mason and Remmel introduced a basis for quasisymmetric functions known as the row-strict quasisymmetric Schur functions. This basis is generated combinatorially by fillings of composition diagrams that are analogous to the row-strict tableaux that generate Schur functions. We introduce a modification known as Young row-strict quasisymmetric Schur functions, which are generated by row-strict Young composition fillings. After discussing basic combinatorial properties of these functions, we define a skew Young row-strict quasisymmetric Schur function using the Hopf algebra of quasisymmetric functions and then prove this is equivalent to a combinatorial description. We also provide a decomposition of the skew Young row-strict quasisymmetric Schur functions into a sum of Gessel’s fundamental quasisymmetric functions and prove a multiplication rule for the product of a Young row-strict quasisymmetric Schur function and a Schur function