Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions

A Factorization Theorem for Classical Group Characters, with Applications to Plane Partitions and Rhombus Tilings

1 , . . . , xn, x −1 n factorizes into a product of two odd orthogonal characters of rectangular shape, one of which is evaluated at −x1, . . . ,−xn, if M is even, while it factorizes into a product of a symplectic character and an even orthogonal character, both of rectangular shape, if M is odd. It is furthermore shown that the first factorization implies a factorization theorem for rhombus t...

Alternating Sign Matrices and Descending Plane Partitions

An alternating sign matrix is a square matrix such that (i) all entries are 1,-1, or 0, (ii) every row and column has sum 1, and (iii) in every row and column the nonzero entries alternate in sign. Striking numerical evidence of a connection between these matrices and the descending plane partitions introduced by Andrews (Invent. Math. 53 (1979), 193-225) have been discovered, but attempts to p...

Totally Symmetric Self-Complementary Plane Partitions and Alternating Sign Matrices

We present multiresidue integral formulae for partial sums in the basis of link patterns of the polynomial solution to the level 1 Uq(ŝl2) quantum Knizhnik–Zamolodchikov equation at generic values of the quantum parameter q. These allow for rewriting and generalizing a recent conjecture [Di Francesco ’06] connecting the above to generating polynomials for weighted Totally Symmetric Self-Complem...

A Protobijection between Alternating Sign Matrices and Descending Plane Partitions

We construct a direct natural bijection between descending plane partitions without any special part and permutations. The directness is in the sense that the bijection avoids any reference to nonintersecting lattice paths. The advantage of the bijection is that it provides an interpretation for the seemingly long list of conditions needed to define descending plane partitions. Unfortunately, t...

A polynomial method for the enumeration of plane partitions and alternating sign matrices