论文信息 - Product evaluations of lefschetz determinants for grassmannians and of determinants of multinomial coefficients

Product evaluations of lefschetz determinants for grassmannians and of determinants of multinomial coefficients

Abstract A general result which produces product evaluations of determinants of certain raising operators for sl(2) representations is obtained. The most combinatorially interesting cases occur for self-dual raising operators of Peck posets. Applications include the following: A nice product expression is found for the determinant of the Lefschetz duality linear transformation on the cohomology of a Grassmannian. Known product expressions for the cardinalities of two sets of plane partitions are re-derived. The appearance of rising factorials for the hooks in one of these product expressions is “explained” by the appearance of rising factorials in sl(2) determinants. A higher dimensional generalization in a certain sense of MacMahon's famous product enumeration result for Ferrers diagrams contained in a box is stated in the context of nonintersecting lattice paths.

Robert A. Proctor

[1] R. Stanley. Theory and Application of Plane Partitions. Part 2 , 1971 .

[2] Robert A. Proctor. Solution of a sperner conjecture of stanley with a construction of gelfand , 1990, J. Comb. Theory A.

[3] Robert A. Proctor. Representations of $\mathfrak{sl}( 2,\mathbb{C} )$ on Posets and the Sperner Property , 1982 .

[4] J. Humphreys. Introduction to Lie Algebras and Representation Theory , 1973 .

[5] I. G. MacDonald,et al. Symmetric functions and Hall polynomials , 1979 .

[6] Richard P. Stanley,et al. Weyl Groups, the Hard Lefschetz Theorem, and the Sperner Property , 1980, SIAM J. Algebraic Discret. Methods.

[7] Robert A. Proctor. Odd symplectic groups , 1988 .

[8] C. Krattenthaler. A determinant evaluation and some enumeration results for plane partitions , 1990 .

[9] Richard P. Stanley,et al. Quotients of peck posets , 1984 .

[10] Jerrold R. Griggs,et al. On Chains and Sperner k-Families in Ranked Posets, II , 1980, J. Comb. Theory, Ser. A.