What is Schur positivity and how common is it?

This is a short note about Schur positivity. We introduce Schur polynomials and explain how they appear in the representation theory of the general linear group. We end with a new result of the author with F. Bergeron and V. Reiner that gives the probability that a homogeneous symmetric polynomial with positive coefficients is Schur positive.Comment: 6 page

Promotion on Generalized Oscillating Tableaux and Web Rotation

We introduce the notion of a generalized oscillating tableau and define a promotion operation on such tableaux that generalizes the classical promotion operation on standard Young tableaux. As our main application, we show that this promotion corresponds to rotation of the irreducible $A_2$-webs of G. Kuperberg.Comment: Comments welcom

Antipode formulas for some combinatorial Hopf algebras

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic analogues of the symmetric functions, quasisymmetric functions, and noncommutative symmetric functions.Comment: 26 page

The probability of positivity in symmetric and quasisymmetric functions

Given an element in a finite-dimensional real vector space, $V$, that is a nonnegative linear combination of basis vectors for some basis $B$, we compute the probability that it is furthermore a nonnegative linear combination of basis vectors for a second basis, $A$. We then apply this general result to combinatorially compute the probability that a symmetric function is Schur-positive (recovering the recent result of Bergeron--Patrias--Reiner), $e$-positive or $h$-positive. Similarly we compute the probability that a quasisymmetric function is quasisymmetric Schur-positive or fundamental-positive. In every case we conclude that the probability tends to zero as the degree of a function tends to infinity

Minuscule reverse plane partitions via quiver representations

A nilpotent endomorphism of a quiver representation induces a linear transformation on the vector space at each vertex. Generically among all nilpotent endomorphisms, there is a well-defined Jordan form for these linear transformations, which is an interesting new invariant of a quiver representation. If $Q$ is a Dynkin quiver and $m$ is a minuscule vertex, we show that representations consisting of direct sums of indecomposable representations all including $m$ in their support, the category of which we denote by $\mathcal{C}_{Q,m}$, are determined up to isomorphism by this invariant. We use this invariant to define a bijection from isomorphism classes of representations in $\mathcal{C}_{Q,m}$ to reverse plane partitions whose shape is the minuscule poset corresponding to $Q$ and $m$. By relating the piecewise-linear promotion action on reverse plane partitions to Auslander-Reiten translation in the derived category, we give a uniform proof that the order of promotion equals the Coxeter number. In type $A_n$, we show that special cases of our bijection include the Robinson-Schensted-Knuth and Hillman-Grassl correspondences.Comment: Comments welcom

Path-cordial abelian groups

A labeling of the vertices of a graph by elements of any abelian group A induces a labeling of the edges by summing the labels of their endpoints. Hovey defined the graph G to be A-cordial if it has such a labeling where the vertex labels and the edge labels are both evenly-distributed over A in a technical sense. His conjecture that all trees T are A-cordial for all cyclic groups A remains wide open, despite significant attention. Curiously, there has been very little study of whether Hovey’s conjecture might extend beyond the class of cyclic groups. We initiate this study by analyzing the larger class of finite abelian groups A such that all path graphs are A-cordial. We conjecture a complete characterization of such groups, and establish this conjecture for various infinite families of groups as well as for all groups of small order