For a semisimple Lie algebra g the orbit method attempts to assign representations of g to (coadjoint) orbits in g∗. Orbital varieties are particular Lagrangian subvarieties of such orbits leading to highest weight representations of g. In sln orbital varieties are described by Young tableaux. Inclusion relation on orbital variety closures defines a partial order on Young tableaux. Our aim is t...

The two parameter family of coefficients Kλμ(q, t) introduced by Macdonald are conjectured to (q, t) count the standard tableaux of shape λ. If this conjecture is correct, then there exist statistics aμ(T ) and bμ(T ) such that the family of symmetric functions Hμ[X; q, t] = ∑ λKλμ(q, t)sλ[X] are generating functions for the standard tableaux of size |μ| in the sense that Hμ[X; q, t] = ∑ T q aμ...

We derive combinatorial identities, involving the Bernoulli and Euler numbers, for the numbers of standard Young tableaux of certain skew shapes. This generalizes the classical formulas of D. André on the number of up-down permutations. The analysis uses a transfer operator approach extending the method of Elkies, combined with an identity expressing the volume of a certain polytope in terms of...

In this paper the number of standard Young tableaux (SYT) is evaluated by the methods of multiple integrals and combinatorial summations. We obtain the product formulas of the numbers of skew SYT of certain truncated shapes, including the skew SYT ((n + k)r+1, nm−1)/(n − 1)r truncated by a rectangle or nearly a rectangle, the skew SYT of truncated shape ((n+ 1)3, nm−2)/(n− 2)\ (22), and the SYT...

We exhibit a weight-preserving bijection between semi-standard Young tableaux and semi-skyline augmented fillings to provide a combinatorial proof that the Schur functions decompose into nonsymmetric functions indexed by compositions. The insertion procedure involved in the proof leads to an analogue of the Robinson-SchenstedKnuth Algorithm for semi-skyline augmented fillings. This procedure co...

This paper contains two results on the number fσ/τ of standard skew Young tableaux of shape σ/τ . The first concerns generating functions for certain classes of “periodic” shapes related to work of Gessel-Viennot and Baryshnikov-Romik. The second result gives an evaluation of the skew Schur function sλ/μ(x) at x = (1, 1/22k , 1/32k , . . . ) for k = 1, 2, 3 in terms of fσ/τ for a certain skew s...