We continue the study of the rational-slope generalized q,t-Catalan numbers c m,n (q,t). We describe generalizations of the bijective constructions of J. Haglund and N. Loehr and use them to prove a weak symmetry property c m,n (q,1)=c m,n (1,q) for m=kn±1. We give a bijective proof of the full symmetry c m,n (q,t)=c m,n (t,q) for min(m,n)≤3. As a corollary of these combinatorial constructions, we give a simple formula for the Poincaré polynomials of compactified Jacobians of plane curve singularities x kn±1=y n . We also give a geometric interpretation of a relation between rational-slope Catalan numbers and the theory of (m,n)-cores discovered by J. Anderson.
We study geometric and combinatorial properties of the degenerate flag varieties of type A. These varieties are acted upon by the automorphism group of a certain representation of a type A quiver, containing a maximal torus T . Using the group action, we describe the moment graphs, encoding the zero- and one-dimensional T -orbits. We also study the smooth and singular loci of the degenerate flag varieties. We show that the Euler characteristic of the smooth locus is equal to the large Schröder number and the Poincaré polynomial is given by a natural statistics counting the number of diagonal steps in a Schröder path. As an application we obtain a new combinatorial description of the large and small Schröder numbers and their q-analogues.
The classical matrix-tree theorem discovered by G.Kirchhoff in 1847 expresses the principal minor of the (n x n) Laplace matrix as a sum of monomials of matrix elements indexed by directed trees with n vertices. We prove, for any k >= n, a three-parameter family of identities between degree k polynomials of matrix elements of the Laplace matrix. For k=n and special values of the parameters the identity turns to be the matrix-tree theorem.
For the same values of parameters and arbitrary k >= n the left-hand side of the identity becomes a specific polynomial of the matrix elements called higher determinant of the matrix. We study properties of the higher determinants; in particular, they have an application (due to M.Polyak) in the topology of 3-manifolds.
FFLV polytopes describe monomial bases in irreducible representations of (Formula presented.) and (Formula presented.). We study various sets of vertices of FFLV polytopes. First, we consider the special linear case. We prove the locality of the set of vertices with respect to the type A Dynkin diagram. Then we describe all the permutation vertices and after that we describe all the simple vertices and prove that their number is equal to the large Schröder number. Finally, we derive analogous results for symplectic Lie algebras.