Published in Journal of Algebraic Combinatorics
As pointed out in Arbarello and Cornalba (J. Alg. Geom.5 (1996), 705–749), a theorem due to Di Francesco, Itzykson, and Zuber (see Di Francesco, Itzykson, and Zuber, Commun. Math. Phys.151 (1993), 193–219) should yield new relations among cohomology classes of the moduli space of pointed curves. The coefficients appearing in these new relations can...
Published in Journal of Algebraic Combinatorics
A t-cover of a quadric \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$Q$$ \end{document} is a set C of t-dimensional subspaces contained in \documentclass[12pt]{minim...
Published in Journal of Algebraic Combinatorics
We introduce the notion of 321-avoiding permutations in the affine Weyl group W of type An − 1 by considering the group as a George group (in the sense of Eriksson and Eriksson). This enables us to generalize a result of Billey, Jockusch and Stanley to show that the 321-avoiding permutations in W coincide with the set of fully commutative elements;...
Published in Journal of Algebraic Combinatorics
Let Xn be either the symmetric group on n letters, the set of planar binary n-trees or the set of vertices of the (n − 1)-dimensional cube. In each case there exists a graded associative product on ⊕n≥0K[Xn]. We prove that it can be described explicitly by using the weak Bruhat order on Sn, the left-to-right order on planar trees, the lexicographic...
Published in Journal of Algebraic Combinatorics
Let Γ denote a bipartite distance-regular graph with diameter D ≥ 3 and valency k ≥ 3. Suppose θ0, θ1, ..., θD is a Q-polynomial ordering of the eigenvalues of Γ. This sequence is known to satisfy the recurrence θi − 1 − βθi + θi + 1 = 0 (0 > i > D), for some real scalar β. Let q denote a complex scalar such that q + q−1 = β. Bannai and Ito have co...
Published in Journal of Algebraic Combinatorics
Answering a question of Kuniba, Misra, Okado, Takagi, and Uchiyama, it is shown that certain higher level Demazure characters of affine type A, coincide with the graded characters of coordinate rings of closures of conjugacy classes of nilpotent matrices.
Published in Journal of Algebraic Combinatorics
If a GQ S′ of order (s, s) is contained in a GQ S of order (s, s2) as a subquadrangle, then for each point X of S\S′ the set of points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \be...
Published in Journal of Algebraic Combinatorics
When can one see from the spectrum of a graph whether it is distance-regular or not? We give some new results for when this is the case. As a consequence we find (among others) that the following distance-regular graphs are uniquely determined by their spectrum: The collinearity graphs of the generalized octagons of order (2,1), (3,1) and (4,1), th...
Published in Journal of Algebraic Combinatorics
Let Γ be a distance-regular graph with adjacency matrix A. Let I be the identity matrix and J the all-1 matrix. Let p be a prime. We study the p-rank of the matrices A + bJ − cI for integral b, c and the p-rank of corresponding matrices of graphs cospectral with Γ. Using the minimal polynomial of A and the theory of Smith normal forms we first dete...
Published in Journal of Algebraic Combinatorics
Given a power series, the coefficients of the formal inverse may be expressed as polynomials in the coefficients of the original series. Further, these polynomials may be parameterized by certain ordered, labeled forests. There is a known formula for the formal inverse, which indirectly counts these classes of forests, developed in a non-direct man...
