We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal’s category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatori...

We analyze the structure of the Malvenuto-Reutenauer Hopf algebraSSym of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration, and show that it decomposes as a crossed product over the Hopf algebra of quasi-symmetric functions. In addition, we describe the structure constants of the...

We describe recent work of Klyachko, Totaro, Knutson, and Tao that characterizes eigenvalues of sums of Hermitian matrices and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Grassmann varieties, and singular values of sums and products of arbitrary matrices. Recent breakthroughs, prima...

An n-core partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously a-core and b-core for two relatively prime integers a and b, which correspond to abacus diagrams and the combinatorics of the affine symmetric group (type A). We observe that self-conjugate simultaneous core partitions correspond to type C combinato...

A compact symmetric space, for purposes of this article, is a quotient G/K, where G is a compact connected Lie group and K is the identity component of the subgroup of fixed points of an involution. A branching theorem describes how an irreducible representation decomposes upon restriction to a subgroup. The article deals with branching theorems for the passage from G to K2 ×K1, where G/(K2 ×K1...

We give a direct proof of the equivalence between the Giambelli and Pieri type formulas for Hall-Littlewood functions using Young’s raising operators, parallel to joint work with Buch and Kresch for the Schubert classes on isotropic Grassmannians. We prove several closely related mirror identities enjoyed by the Giambelli polynomials, which lead to new recursions for Schubert classes. The raisi...

Many combinatorial Hopf algebras H in the literature are the functorial image of a linearized Hopf monoid H. That is, H = K(H) or H = K(H). Unlike the functor K, the functor K applied to H may not preserve the antipode of H. In this case, one needs to consider the larger Hopf monoid L×H to getH = K(H) = K(L×H) and study the antipode in L × H. One of the main results in this paper provides a can...

Let X be an orthogonal Grassmannian parametrizing isotropic subspaces in an even dimensional vector space equipped with a nondegenerate symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in the classical and quantum cohomology ring of X as a polynomial in certain special Schubert classes. Our analysis reveals a surprising relation between the Schubert calcu...

Let X be a symplectic or odd orthogonal Grassmannian which parametrizes isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove quantum Giambelli formulas which express an arbitrary Schubert class in the small quantum cohomology ring of X as a polynomial in certain special Schubert classes, extending the authors’ cohomological Giambelli formulas. 0. I...