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We start with a ( q, t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is the order of a finite field. These ( q, t)-binomial coefficients and their interpretations generalize further in two directions, one relating to column-strict tableaux and Macdonald’s “7 th variation” of Schur functions, the other relating to permutation statistics and Hilbert series from the invariant theory of $GL_{n}({\mathbb{F}}_{q})$ . [ABSTRACT FROM AUTHOR]

Monotone triangles are a rich extension of permutations that are in bijection with alternating sign matrices. The notions of weak order and descent sets for permutations are generalized here to monotone triangles, and shown to enjoy many analogous properties. It is shown that any linear extension of the weak order gives rise to a shelling order on a poset, recently introduced by Terwilliger, whose maximal chains are in bijection with monotone triangles; among these shellings is a family of EL-shellings. The weak order turns out to encode an action of the 0-Hecke monoid of type A on the monotone triangles, generalizing the usual bubble-sorting action on permutations. It also leads to a notion of descent set for monotone triangles, having another natural property: the surjective algebra map from the Malvenuto–Reutenauer Hopf algebra of permutations into quasisymmetric functions extends in a natural way to an algebra map out of the recently-defined Cheballah–Giraudo–Maurice algebra of alternating sign matrices. [ABSTRACT FROM AUTHOR]

We show that the Koszul homology algebra of the second Veronese subalgebra of a polynomial ring over a field of characteristic zero is generated, as an algebra, by the homology classes corresponding to the syzygies of the lowest linear strand. [ABSTRACT FROM AUTHOR]

Working in a polynomial ring S=k[x1,...,xn]$S={\mathbf {k}}[x_1,\ldots ,x_n]$, where k${\mathbf {k}}$ is an arbitrary commutative ring with 1, we consider the d$d$th Veronese subalgebras R=S(d)$R={S^{(d)}}$, as well as natural R$R$‐submodules M=S(⩾r,d)$M={S^{({\geqslant r},{d})}}$ inside S$S$. We develop and use characteristic‐free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple GLn(k)$GL_n({\mathbf {k}})$‐equivariant minimal free R$R$‐resolutions for the quotient ring k=R/R+${\mathbf {k}}=R/R_+$ and for these modules M$M$. These also lead to elegant descriptions of ToriR(M,M′)$\operatorname{Tor}^R_i(M,M^{\prime})$ for all i$i$ and HomR(M,M′)$\operatorname{Hom}_R(M,M^{\prime})$ for any pair of these modules M,M′$M,M^{\prime}$. [ABSTRACT FROM AUTHOR]

We define toric partial orders, corresponding to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets correspond to finite posets under the equivalence relation generated by converting minimal elements into maximal elements, or sources into sinks. We derive toric analogues for several features of ordinary partial orders, such as chains, antichains, transitivity, Hasse diagrams, linear extensions, and total orders. [ABSTRACT FROM AUTHOR]

Abstract: A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant: [ ] defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients; [ ] is a multivariate generating function for integer weight vectors that give minimum total weight to a unique base of the matroid; [ ] is equivalent, via the Hopf antipode, to a generating function for integer weight vectors which keeps track of how many bases minimize the total weight; [ ] behaves simply under matroid duality; [ ] has a simple expansion in terms of -partition enumerators; [ ] is a valuation on decompositions of matroid base polytopes. This last property leads to an interesting application: it can sometimes be used to prove that a matroid base polytope has no decompositions into smaller matroid base polytopes. Existence of such decompositions is a subtle issue arising from the work of Lafforgue, where lack of such a decomposition implies that the matroid has only a finite number of realizations up to scalings of vectors and overall change-of-basis. [Copyright &y& Elsevier]

The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of a line graph. The first two results interact extremely well with the third. For example, they imply that in a regular nonbipartite graph, the critical group of the graph and that of its line graph determine each other uniquely in a simple fashion. [ABSTRACT FROM AUTHOR]