The Cascade Lectures in Combinatorics is a series of combinatorial workshops held each fall and each spring on a Saturday in the Pacific Northwest Region (or sometimes on Zoom), funded by the National Science Foundation.  The workshop this fall, the 5th installment of the Cascade Lectures in Combinatorics, will be held as an online meeting, held on Zoom, on Saturday, November 4, 2023.  It will include four one hour invited talks as well as online social gatherings in the form of coffee breaks and a lunch break. Additional information is on the main CALICO web site at https://pages.uoregon.edu/plhersh/CALICO/ 

Talk titles and abstracts:

Chris Eur: A Tale of two rings

Abstract: A complex projective manifold carries two well-studied rings, namely, the cohomology ring and the Grothendieck K-ring of vector bundles. For toric varieties, these have polyhedral descriptions, as the polytope algebra and the algebra of piecewise polynomials. For special toric varieties, we show an exceptional isomorphism between these two rings, different from the classical Hirzebruch-Riemann-Roch theorem, and discuss its utility in combinatorial contexts.

Joint works with Andrew Berget, Alex Fink, June Huh, Matt Larson, Hunter Spink, and Dennis Tseng.

Sergey Fomin: Incidences and tilings

Abstract: We show that various classical theorems of real/complex linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a single "master theorem" that involves an arbitrary tiling of a closed oriented surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and generalizing the known ones.

This is joint work with Pavlo Pylyavskyy.

Maria M. Gillespie: Battery-powered tableaux, Springer theory, and the Delta conjecture

Abstract: We present new formulas for the t=0 specialization of the polynomials involved in the Delta conjecture. One is a combinatorial Schur expansion in terms of "battery-powered tableaux", and its related formulation is simply an adjoint Schur operator applied to a Hall-Littlewood polynomial. This generalizes to give formulas for the Frobenius character of the cohomology ring of the "Delta-Springer varieties" defined by Griffin, Levinson, and Woo, which fit into the generalized Springer theory of Borho and MacPherson. We will describe how this generalized theory of partial resolutions of nilpotent varieties leads to our results, and state some more general conjectural formulas towards a Schur expansion for the Delta conjecture at the end.

This is joint work with Sean Griffin.

Sheila Sundaram: Stirling representations, supersolvable matroids and Koszul duality

Abstract: (Joint work with Ayah Almousa and Vic Reiner)

The unsigned Stirling numbers c(n,k) of the first kind give the Hilbert function for two algebras associated to the hyperplane arrangement in type A, the Orlik-Solomon algebra and the graded Varchenko-Gelfand algebra. Both algebras carry symmetric group actions with a rich structure, and have been well studied by topologists, algebraists and combinatorialists: the first coincides with the Whitney homology of the partition lattice, and the second with a well known decomposition (Thrall's decomposition, giving the higher Lie characters) of the universal enveloping algebra of the free Lie algebra. In each case the graded representations have dimension c(n,k).

Both these algebras are examples of Koszul algebras, for which the Priddy resolution defines a group-equivariant dual Koszul algebra. Now the Hilbert function is given by the Stirling numbers S(n,k) of the second kind, and hence the Koszul duality relation defines representations of the symmetric group whose dimensions are the numbers S(n,k). Investigating this observation led to the realisation that this situation generalises to all supersolvable matroids. The Koszul duality recurrence is shown to have interesting consequences.

For the resulting group representations, it implies the existence of branching rules which, in the case of the braid arrangement, specialise by dimension to the classical enumerative recurrences satisfied by the Stirling numbers of both kinds. It also implies representation stability in the sense of Church and Farb. The associated Koszul dual representations appear to have other properties that are more mysterious; for example, in the case of the braid arrangement, the Stirling representations of the Koszul dual are sometimes tantalisingly close to being permutation modules. I will endeavour to give a flavour of these phenomena in the talk.