VT Algebra Seminar — Spring 2021

In 2015, Brosnan and Chow, and independently Guay-Paquet, proved the Shareshian--Wachs conjecture, which links the combinatorics of chromatic symmetric functions to the geometry of Hessenberg varieties via a permutation group action on the cohomology ring of regular semisimple Hessenberg varieties. This talk will give a brief overview of that story and discuss how the dot action can be computed in all Lie types using the Betti numbers of certain nilpotent Hessenberg varieties. As an application, we obtain new geometric insight into certain linear relations satisfied by chromatic symmetric functions, known as the modular law. This is joint work with Eric Sommers.

For a complex reductive group $G$, classical Steinberg theory is based on the fact that the action of $G$ on the double flag variety $G/B\times G/B$ has a finite number of orbits, parametrized by the Weyl group $W$. Relying on the moment map corresponding to this action, the conormal bundle of each orbit is mapped to a nilpotent variety in $Lie(G)$. Therefore, one gets a map from the Weyl group to the set of nilpotent orbits, often called the Steinberg map. In type A, this map can be computed explicitly in terms of classical combinatorial algorithms, namely the Robinson-Schensted correspondence. In this talk, we consider a double flag variety $G/P\times K/Q$ associated to a symmetric pair $(G,K)$. Following Steinberg's approach, we define two maps to the set of nilpotent $K$-orbits of $Lie(K)$ and of its Cartan complement, respectively. We focus on type AIII, the orbits of the double flag variety are then parametrized by a set of pairs of partial permutations. We compute the two maps, by relying on a combinatorial procedure that extends the classical Robinson-Schensted correspondence. This talk is based on a joint work with Kyo Nishiyama.

The quantum cohomology ring of the Grassmannian is completely determined by the Pieri rule for multiplying by Schubert classes indexed by row or column-shaped partitions. In this talk, we will explain an equivariant generalization of Postnikov’s quantum Pieri rule in terms of cylindric shapes. Unlike earlier equivariant quantum Pieri and Littlewood-Richardson rules, this formula does not require any calculations in a different Grassmannian or two-step flag variety.

We define a class of transversal slices in manifolds which are quasi-Poisson for the action of a complex semisimple group G. This is a multiplicative analogue of Whittaker reduction. One example is the multiplicative universal centralizer Z of G, which inherits a symplectic structure in this way. We construct a smooth partial compactification of Z by taking the closure of each centralizer fiber in the wonderful compactification of G. We show that the induced Poisson structure on the compactification is log-symplectic, and that Z is its open dense symplectic leaf.

The quantum cohomology ring of the Grassmannian is completely determined by the Pieri rule for multiplying by Schubert classes indexed by row or column-shaped partitions. In this talk, we will explain an equivariant generalization of Postnikov’s quantum Pieri rule in terms of cylindric shapes. Unlike earlier equivariant quantum Pieri and Littlewood-Richardson rules, this formula does not require any calculations in a different Grassmannian or two-step flag variety.