The UC Berkeley Combinatorics Seminar

Fall 2024 - Wednesdays 11:10am - 12:00pm, Evans 891
Introductory pre-talk for graduate students (open to all) 10:40am - 11:05am, Evans 891
Zoom Meeting ID: 953 1397 4237, the password is the name of our favorite combinatorial sequence
Organizers: Christian Gaetz, Nicolle González, Mitsuki Hanada, and John Lentfer

If you would like to be added to the seminar mailing list, contact Nicolle González.
If you would like to view recordings of some of the talks (in a Google drive), contact Mitsuki Hanada for access.

Conference Announcement: the Fall 2024 BAD Math Day with be held at UC Berkeley on December 7th, 2024.

DATE SPEAKER TITLE (click to show abstract)
August 28th (No pre-talk) Lauren Williams, Harvard
Cyclic partial orders, Parke-Taylor identities, and the magic number conjecture for the m=2 amplituhedronThe magic number conjecture says that the cardinality of a tiling of the amplituhedron An,k,m is the number of plane partitions which fit inside a k by (n-k-m) by m/2 box. (This is a generalization of the fact that triangulations of even-dimensional cyclic polytopes have the same size.) I'll explain how we prove the magic number conjecture for the m=2 amplituhedron; we also show that all positroid tilings of the hypersimplex have the same cardinality. Along the way, we give volume formulas for Parke-Taylor polytopes in terms of circular extensions of cyclic partial orders, and we prove new variants of the classical Parke-Taylor identities. This is joint work with Matteo Parisi, Melissa Sherman-Bennett, and Ran Tessler.
September 4th Christian Gaetz, UC Berkeley
Hypercube decompositions and combinatorial invariance for Kazhdan-Lusztig polynomialsKazhdan-Lusztig polynomials are of foundational importance in geometric representation theory. Yet the Combinatorial Invariance Conjecture, due to Lusztig and to Dyer, suggests that they only depend on the combinatorics of Bruhat order. I'll describe joint work with Grant Barkley in which we adapt the hypercube decompositions introduced by Blundell-Buesing-Davies-Veličković-Williamson to prove this conjecture for Kazhdan-Lusztig R-polynomials in the case of elementary intervals in the symmetric group. This significantly generalizes the main previously known case of the conjecture, that of lower intervals.
September 11th Joshua Turner, UC Davis
Haiman ideals, link homology, and affine Springer fibersWe will discuss a class of ideals in a polynomial ring studied by Mark Haiman in his work on the Hilbert scheme of points, and ask some purely algebraic questions about them. It turns out that these questions are very closely tied to homology of affine Springer fibers, Khovanov-Rozansky homology of links, and to the ORS conjecture. We will discuss which cases are known and unknown, and compute some simple examples.
September 18th Tonie Scroggin, UC Davis
Splicing Positroid VarietiesGalashin and Lam established the connection between the torus equivariant cohomology on $\Pi_{k,n}^\circ$ and Khovanov-Rozansky homology HHH of the torus link $T(k,n-k)$. Additionally, the multiplication of braids $T(k,s)\cdot T(k,t)\to T(k,s+t)$ suggests that there should be a map of open positroid varieties. In this talk, I will describe an explicit isomorphism which decomposes an open subset in the open positroid variety $\Pi_{k,n}^{\circ}$ in the Grassmannian $\mathrm{Gr}(k,n)$ into the product of two open positroid varieties $\Pi_{k,n-a+1}^{\circ}\times \Pi_{k,a+k-1}^{\circ}$. The isomorphism is given by freezing a certain subset of cluster variables in $\Pi_{k,n}^\circ$ and constructing a cluster quasi-homomorphism. This is joint work with Eugene Gorsky.
September 25th John Lentfer, UC Berkeley
The (1,2)-bosonic-fermionic coinvariant ringIn 1994 Haiman introduced the ring of diagonal coinvariants, which is a quotient of a polynomial ring in two sets of commutative variables by invariants of the diagonal action of the symmetric group. Recently, there has been much interest in studying a more general class of coinvariant rings with k sets of n commutative (bosonic) variables and j sets of n anticommutative (fermionic) variables; denote this ring by R_n^{(k,j)}. We will focus on the coinvariant ring R_n^{(1,2)}, with one set of bosonic and two sets of fermionic variables. By interpolating between the modified Motzkin path basis for R_n^{(0,2)} of Kim--Rhoades (2022) and the super-Artin basis for R_n^{(1,1)} conjectured by Sagan--Swanson (2024) and proven by Angarone et al. (2024), we propose a monomial basis for R_n^{(1,2)}. We use the proposed basis to give combinatorial formulas for its conjectural Hilbert series and Frobenius series. We will explain how our work on R_n^{(1,2)} relates to the Theta conjecture and recent work of Iraci, Nadeau, and Vanden Wyngaerd (2023). In the pre-talk, we will see examples of coinvariant algebras, and see how to compute their Hilbert and Frobenius series.
October 2nd Matthias Beck, SFSU
Acyclotopes and tocylotopesThere is a well-established dictionary between zonotopes, hyperplane arrangements, and (oriented) matroids. Arguably one of the most famous examples is the class of graphical zonotopes, also called acyclotopes, which encode subzonotopes of the type-A root polytope, the permutahedron. Stanley (1991) gave a general interpretation of the coefficients of the Ehrhart polynomial (integer-point counting function for a polytope) of a zonotope via linearly independent subsets of its generators. Applying this to the graphical case shows that Ehrhart coefficients count induced forests of the graph of fixed sizes. Our first goal is to extend and popularize this story to other root systems, which on the combinatorial side is encoded by signed graphs analogously to the work by Greene and Zaslavsky (1983). We compute the Ehrhart polynomial of the acyclotope in the signed case, and we give a matroid-dual construction, giving rise to tocyclotopes, and compute their Ehrhart polynomials. Applying the same duality construction to a general integral matrix gives rise to a lattice Gale zonotope, whose combinatorial structure was studied by McMullen (1971) and D'Adderio-Luca (2012). We describe its Ehrhart polynomials in terms of the given matrix. This is joint work with Eleonore Bach and Sophie Rehberg.
October 9th David Eisenbud, UC Berkeley
Numerical semigroups, Weierstrass points and the Kunz conesPretalk: The Riemann-Roch theorem for Riemann surfaces: what it is, and why we care. Abstract: A numerical semigroup is a subset of the non-negative integers with finite complememt, closed under addition (and thus including 0). Such things arise in many contexts, including combinatorics, commutative algebra, and the theory of Riemann surfaces. I'll focus on two areas: what is known about the Weierstrass semigroups of Riemann surfaces (I'll give the definitions!); and the scheme invented by Ernst Kunz to classify numerical semigroups with a given smallest non-zero element. I'll also state an old (and still open) problem of Vladimir Arnol'd related to the classification of slightly more general objects.
October 16th Mitsuki Hanada, UC Berkeley
A charge monomial basis of the Garsia-Procesi ring The two well-known monomial bases of the (classical) coinvariant ring, the Artin basis and the descent basis, are indexed by permutations and correspond to the permutation statistics inv and maj respectively. These bases are compatible with the Hilbert series of the coinvariant ring, which is $[n]_q!$. We construct a basis of the Garsia-Procesi ring $R_\mu$, a quotient of the coinvariant ring, which is compatible with its Hilbert series in the same manner. This basis coincides with the basis defined by Carlsson-Chou (2024). Our new construction connects the combinatorics of the basis with the combinatorial formula for the modified Hall-Littlewood polynomials $\tilde{H}_\mu[X;q]$, due to Lascoux, which expresses the polynomials as a sum over standard tableaux that satisfy a catabolizability condition. We use this construction to give an elementary proof of the fact that the graded Frobenius character of $R_{\mu}$ is given by the catabolizability formula for $\tilde{H}_\mu[X;q]$ In the pretalk, we will discuss the combinatorial tools needed to compute $\tilde{H}_\mu[X;q]$.
October 23rd Hannah Friedman, UC Berkeley
Likelihood Geometry of the Squared Grassmannian Determinantal point processes are a family well-studied models arising in physics and machine learning. I will talk about recent work on maximum likelihood estimation for determinantal point processes from an algebraic perspective. There is a nice connection between determinantal point processes and the squared Grassmannian, which we use to show that the complexity of maximum likelihood estimation (ML degree) grows exponentially using topology and combinatorics.
October 30th Michelle Wachs, University of Miami
On an $n$-ary generalization of the Lie representation and tree Specht modulesThe Filippov n-algebra is a natural n-ary generalization of Lie algebra that is of interest in elementary particle physics. It is also of interest in combinatorics because it yields representations of the symmetric group that generalize the well studied Lie representation. Our ultimate aim is to determine the multiplicities of the irreducible representations in the representation of the symmetric group on the multilinear component of the free Filippov n-algebra with k brackets. This had been done for the ordinary Lie representation (n=2 case) by Kraskiewicz and Weyman. The k=2 case was handled in work with Friedmann, Hanlon, and Stanley. I will talk on continuing progress for general (n,k) obtained very recently with Friedman and Hanlon. Our main result shows that the multiplicities stabilize in a certain sense when $n$ exceeds $k$. As an important tool in proving this, we present two types of generalizations of Specht module involving trees.
November 6th Rebecca Whitman, UC Berkeley
Identifying graphs by degree sequenceA graph with degree sequence d is a unigraph if it is isomorphic to every graph with degree sequence d. The class of unigraphs can be characterized by degree sequence, but it is not hereditary. I will "improve" it to the smallest hereditary class containing all unigraphs, which we call the hereditary closure of the unigraphs (HCU). The class HCU is then hereditary, characterizable by degree sequences, and invariant under a particular graph decomposition operation. Furthermore, given a class A of graphs, we define the class of A-unigraphs to be graphs identifiable from degree sequence and membership in A. While these classes are often not hereditary, we provide characterizations of the largest hereditary subclass contained in the bipartite-unigraphs, the $k$-partite unigraphs, the perfect-unigraphs, and the chordal-unigraphs. This is partially joint work with Michael Barrus and Ann Trenk.
November 13th Monica Vazirani , UC Davis
Skeins on Tori We study skeins on the 2-torus and 3-torus via the representation theory of the double affine Hecke algebra (DAHA) of type A and its connection to quantum D-modules. As an application we can compute the dimension of the generic $SL_N$- and $GL_N$-skein module of the 3-torus for arbitrary N. One necessary ingredient is understanding that the sign idempotent does not annihilate DAHA modules. We use our understanding of the affine Hecke algebra and the combinatorics of multisegments to prove this. This is joint work with Sam Gunningham and David Jordan.
November 20th Sarah Brauner, Brown
Spectrum of random-to-random shuffling in the Hecke algebraThe eigenvalues of a Markov chain determine its mixing time. In this talk, I will describe a Markov chain on the symmetric group called random-to-random shuffling whose eigenvalues have surprisingly elegant—though mysterious—formulas. In particular, these eigenvalues were shown to be non-negative integers by Dieker and Saliola in 2017, resolving an almost 20 year conjecture. In recent work with Ilani Axelrod-Freed, Judy Chiang, Patricia Commins and Veronica Lang, we generalize random-to-random shuffling to a Markov chain on the Type A Iwahori Hecke algebra, and prove combinatorial expressions for its eigenvalues as a polynomial in q with non-negative integer coefficients. Our methods simplify the existing proof for q=1 by drawing novel connections between random-to-random shuffling and the Jucys-Murphy elements of the Hecke algebra.
November 27th No seminar - Thanksgiving
December 4th Mary Claire Simone, UC Davis
The Immersion Poset on PartitionsWe introduce the immersion poset on partitions. Relations in the immersion poset determine when irreducible polynomial representations of the general linear group form an immersion pair, as defined by Prasad and Raghunathan (2022). We develop injections on semistandard Young tableaux given constraints on the shape of the partition, and present results on immersion relations among hook and two column partitions. We classify maximal elements of certain shapes in the standard immersion poset using the hook length formula. Finally, we prove Schur-positivity of power sum symmetric functions on conjectured lower intervals in the immersion poset, addressing questions posed by Sundaram (2018). This is joint work with Lisa Johnston, David Kenepp, Evuilynn Nguyen, Digjoy Paul, Anne Schilling, and Regina Zhou.
December 11th (No pre-talk) Samira Sahar Jamil, SLMath
Computational aspects of Discrete Cubical Homology Discrete cubical homology was developed as a corresponding homology theory for discrete homotopy theory by Barcelo and coauthors. In this talk, I will provide a bird’s-eye view of the work on discrete cubical homology for simple graphs, highlighting its connections to the discrete homotopy theory of graphs. I will present my recent results aimed at enhancing the computational efficiency of discrete cubical homology on a specialized class of graphs known as c_1-digital images.
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