## The UC Berkeley Combinatorics SeminarSpring 2024 - Wednesdays 2:10pm - 3:00pm, Evans 939Introductory pre-talk for graduate students (open to all) 1:40pm - 2:05pm, Evans 939 Zoom Meeting ID: 953 1397 4237, the password is the name of our favorite combinatorial sequence Organizers: 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. |
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DATE |
SPEAKER |
TITLE (click to show abstract) |

January 17th | Bernd Sturmfels, MPI MiS Leipzig and UC Berkeley | ## Algebraic Varieties in Quantum ChemistryWe discuss the algebra and combinatorics behind coupled cluster (CC) theory of quantum many-body systems. The high-dimensional eigenvalue problems that encode the electronic Schroedinger equation are approximated by polynomial systems at various levels of truncation. The exponential parametrization of the eigenstates gives rise to truncation varieties. These generalize Grassmannians in their Pluecker embedding. We explain how to derive Hamiltonians, we offer a detailed study of truncation varieties and their CC degrees, and we discuss the solution of the CC equations. This is joint work with Fabian Faulstich and Svala Sverrisdóttir. |

January 24th | Foster Tom, MIT | ## A signed $e$-expansion of the chromatic symmetric function and some new $e$-positive graphsWe prove a new signed elementary symmetric function expansion of the chromatic symmetric function of any unit interval graph. We then use sign-reversing involutions to prove new combinatorial formulas for many families of graphs, including the K-chains studied by Gebhard and Sagan, formed by joining cliques at single vertices, and for graphs obtained from them by removing any number of edges from any of the cut vertices. We also introduce a version for the quasisymmetric refinement of Shareshian and Wachs. |

January 31st | Ashleigh Adams, North Dakota State University | ## A Generalized Coinvariant AlgebraThe coinvariant algebra, a polynomial quotient ring whose ideal is S_n-invariant, has been well studied. In this talk we give a natural generalization of this ring motivated by the study of Stanley-Reisner rings, a family of rings that captures the structure of (finite, abstract) simplicial complexes. For the case when the Stanley-Reisner ideal is generated by minimally ordered square-free monomial ideals, we give an algorithm for writing down the Groebner basis. Using this recipe, we describe a combinatorial model for writing down the basis for the generalized coinvariant algebra and then we extend our intuition from these certain cases in order to give an explicit formula for the Hilbert series, in general. We will also give an application for the study of this particular ring. |

February 7th (No pre-talk) |
Danai Deligeorgaki, KTH | ## Colored multiset Eulerian polynomialsThe central objects in this talk are the descent polynomials of colored permutations on multisets, referred to as colored multiset Eulerian polynomials. These polynomials generalize the colored Eulerian polynomials that appear frequently in algebraic combinatorics and are known to admit desirable distributional properties, including real-rootedness, log-concavity, unimodality and the alternatingly increasing property. In joint work with Bin Han and Liam Solus, symmetric colored multiset Eulerian polynomials are identified and used to prove sufficient conditions for a colored multiset Eulerian polynomial to be interlaced by its own reciprocal. This property implies that the polynomial obtains all of the aforementioned distributional properties as well as others, including bi-$\gamma$-positivity. To derive these results, multivariate generalizations of a generating function identity due to MacMahon are deduced. We will also discuss some connections to $s$-Eulerian polynomials and end with open questions. |

February 14th (No pre-talk) |
Svala Sverrisdóttir, UC Berkeley | ## CC degree of the GrassmannianThe CC degree of a variety arises from Quantum chemistry and is the number of complex solutions to a nonlinear eigenvalue problem subjected to that variety. We will determine the CC degree of the Grassmannian of lines in its Plücker embedding, conjectured in earlier work with Fabian Faulstich and Bernd Sturmfels. This rests on the geometry of the graph of a birational parametrization of the Grassmannian. We present a squarefree Gröbner basis for this graph, giving rise to Stanley-Reisner ideals and corresponding simplicial complexes. Additionally we will explore the CC degree of the Grassmannian in general and develop connections to toric degenerations from representation theory. |

February 21st | Rebecca Whitman, UC Berkeley | ## A Hereditary Generalization of Nordhaus-Gaddum GraphsNordhaus and Gaddum proved in 1956 that the sum of the chromatic number Chi of a graph G and its complement is at most |G| + 1. The Nordhaus-Gaddum graphs are the class of graphs satisfying this inequality with equality, and are well-understood. In this talk we consider a hereditary generalization: graphs G for which all induced subgraphs H of G have that the sum of the chromatic numbers of H and its complement is at least |H|. We characterize the forbidden induced subgraphs of this class and find its intersection with a number of common classes, including line graphs. We also discuss Chi-boundedness and algorithmic optimization results. |

February 28th | Joshua Swanson, USC | ## SL(4) web bases from hourglass plabic graphsIn 1995, Kuperberg introduced a remarkable collection of trivalent web bases which encode tensor invariants of SL_3. Extending these bases to general SL_r has been an open problem ever since. We present a solution to the r=4 case by introducing a new generalization of Postnikov's plabic graphs. Joint work with Christian Gaetz, Oliver Pechenik, Stephan Pfannerer, and Jessica Striker. |

March 6th | Peter Samuelson, UC Riverside | ## Symmetric functions and quantum topologyIn the last decade, a number of conjectures about symmetric functions have been proved and/or generalized using the action of the elliptic Hall algebra on Sym. In this talk, we explain how the "Macdonald --> Schur" specialization of this action can be realized by applying constructions from quantum topology to the annulus and torus. We then explain a conjecture that the "type BCD" version of this quantum-topological construction also has a q,t deformation. This deformation would presumably come from a "double affine BMW algebra," which does not seem to exist in the literature.In the pretalk, we will use quantum groups (as a dark-gray box) to motivate the quantum topology constructions that we use in the main talk. We will also do a few concrete computations. |

March 13th | Anthony Licata, Australian National University | ## The combinatorial geometry of spherical objectsThe study of Coxeter groups has a rich interaction with the combinatorics of root systems; in particular, many important questions about Coxeter groups can be answered by carefully studying the action of the group on its associated root lattice. There is a somewhat analogous story for braid groups, where the role of roots is played by spherical objects in an appropriate triangulated category. In this talk I'll try to explain some of the interesting combinatorial and geometric structure - much of it still conjectural - that emerges when one studies these spherical objects. |

March 20th | Zajj Daugherty, Reed College | ## Type C combinatorics of two-pole centralizer algebrasHecke algebras arise in one sense as deformations of Weyl groups, but in another sense as quotients of braid algebras. While some types of braid algebras can only loosely be associated to their namesake, some can be concretely imagined as braid diagrams on k strands, possibly in spaces with one or more punctures. The latter produces a connection to quantum group actions on tensor spaces of particular shapes. For example, the finite Hecke algebra of type A has a natural action on a tensor space V^{\otimes k}, which commutes with the natural action of the quantum group of type A (this is precisely the q-deformation of classical Schur-Weyl duality). Closures of braids then correspond to traces of endomorphisms, giving rise to knot and link invariants via Hecke algebras; actions on tensor space have implications for lattice models in statistical mechanics; and on the connections go. It is from this perspective of Schur-Weyl duality that our story takes place: classifying representations of the type-C affine Hecke algebra was once relegated to geometric methods. However, what one learns from studying two-pole braids, and the corresponding tensor spaces, unlocks a beautiful combinatorial classification of representation theory of the type-C affine Hecke and Temperley-Lieb algebras. |

March 27th | No seminar - Spring break | |

Friday April 5th, 1:40pm-3:00pm, Evans 939 |
Brendon Rhoades, UCSD | ## Increasing subsequences, orbit harmonics, and shadow playThe Schensted correspondence is a bijection between permutations in $S_n$ and pairs of standard Young tableaux $(P,Q)$ with $n$ boxes which have the same shape. This bijection has remarkable properties in algebraic and enumerative combinatorics. Motivated by a problem in cryptography, we study a graded quotient $R_n$ of the polynomial ring in $n \times n$ variables whose standard monomial theory encodes Viennot's shadow line formulation of the Schensted correspondence. The quotient ring $R_n$ may be understood as coming from the locus of permutation matrices via the machine of orbit harmonics. I will report on work of my student Jasper Liu who has extended this theory to colored permutation groups. |

Monday April 8th, 10:40am-12:00pm, Evans 732 |
Christopher Ryba | ## Towards an integral version of the Harish-Chandra theorem for gl_nThe Harish-Chandra theorem tells us that the centre of of the universal enveloping algebra of gl_n (over the complex numbers) is the ring of symmetric polynomials in n variables. In the context of modular representation theory, one might be interested a version of the theorem over the integers or a field of positive characteristic. In this setting, one has to be careful about the precise notion of universal enveloping algebra; we work with the Kostant-Lusztig integral form (rather than the De Concini-Kac integral form). In principle, the centre of the integral form is just some integral form of the symmetric polynomial algebra, but it is subtle, even in the case of gl_2. We explain the structure of the integral form for gl_2. This is joint work with Pablo Boixeda Alvarez. |

April 10th | Andrés Vindas Meléndez, UC Berkeley and Harvey Mudd | ## On the Ehrhart theory of panhandle and paving matroidsWe show that the matroid base polytope P_M of any paving matroid M can be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial of P_M. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of stressed-hyperplane relaxation, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show that they depend only on their design-theoretic parameters. In more recent work, we prove Ehrhart positivity for panhandle matroid polytopes. A standing conjecture posed by Ferroni (2022) asserts that the coefficients of the Ehrhart polynomial of a connected matroid are bounded above by those of the corresponding uniform matroid. We prove Ferroni's conjecture for paving matroids -- a class conjectured to asymptotically contain all matroids. Our proofs rely solely on combinatorial techniques which involve determining intricate interpretations of certain set partitions. |

April 17th | Frank Sottile, Texas A&M | ## A Murnaghan-Nakayama formula in quantum Schubert calculusThe Murnaghan-Nakayama formula expresses the product of a Schur function with a Newton power sum in the basis of Schur functions. An important generalization of Schur functions are Schubert polynomials (both classical and quantum). For these, a Murnaghan-Nakayama formula is geometrically meaningful. In previous work with Morrison, we established a Murnaghan-Nakayama formula for Schubert polynomials and conjectured a quantum version. In this talk, I will discuss some background and then some recent work proving this quantum conjecture. This is joint work with Benedetti, Bergeron, Colmenarejo, and Saliola. |

April 24th (No pre-talk) |
Nicolle González, UC Berkeley | ## Semistandard Parking Functions and a Finite Shuffle TheoremIn this talk I will introduce semistandard parking functions and use those to construct a new family of polynomials, the higher rank (q,t)-Catalan polynomials, that interpolate between (q,t)-Catalan numbers and the Hikita polynomial. I will then show how these polynomials arise naturally in a DAHA-version of the shuffle theorem. This is joint work with Jose Simental and Monica Vazirani. |

Tuesday April 30th, 12:10pm-1:30pm, Evans 939 |
Arun Ram, University of Melbourne | ## ETA Modified Macdonald polynomials are not Macdonald polynomialsI will give a new combinatorial mechanics for computing the plethystic transformation that relates integral form Macdonald polynomials and modified Macdonald polynomials in type A. This new point of view points to a generalisation to all Lie types of modified Macdonald polynomials, integral form Macdonald polynomials and the plethyistic transformation that relates them. Except in type A, there is no immediate relationship to the Macdonald polynomials for all Lie types defined by Macdonald. |

Monday May 6th, 3:10pm-4:30pm, Evans 939 |
Chris Bowman, University of York | ## Meta-Kazhdan—Lusztig combinatorics and Hecke categoriesIn the pretalk we will give a combinatorial introduction to Kazhdan—Lusztig polynomials and attempt to motivate their study. Kazhdan–Lusztig polynomials encode a great deal of character-theoretic and indeed cohomological information about Verma modules. We further know that Kazhdan–Lusztig polynomials often carry information about the radical layers of indecomposable projective and cell modules. Given the almost ridiculous level of detail these polynomials encode, it is natural to ask “what are the limits to what p-Kazhdan–Lusztig combinatorics can tell us about the structure of the Hecke category?” In this talk we discuss recent work towards answering this titillating question. This is joint work with Maud De Visscher, Anton Cox, Alice Del’Arciprete, Amit Hazi, Rob Muth, and Catharina Stroppel. |

May 8th | Liron Speyer, OIST | ## Schurian-infinite blocks of type A Hecke algebrasFor any algebra A over an algebraically closed field F, we say that an A-module M is Schurian if End_A(M) is isomorphic to F. We say that A is Schurian-finite if there are only finitely many isomorphism classes of Schurian A-modules, and Schurian-infinite otherwise. I will present joint work with Susumu Ariki and Sinéad Lyle in which we classified the Schurian-finiteness of blocks of type A Hecke algebras – when e ≠ 2, a block of the Hecke algebra is Schurian-finite if and only if it has finite representation type, if and only it has weight 0 or 1. I will give an overview of the techniques that were used in this project. |