The UC Berkeley Combinatorics Seminar |
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DATE |
SPEAKER |
TITLE (click to show abstract) |

September 5th | Sergei Korotkikh, UC Berkeley | ## A new family of interpolation symmetric functionsDuring 90s several research groups has studied characterizations of symmetric functions in terms of evaluations at certain points. One of the results of that research is the work by Okounkov from 1998, where all symmetric functions satisfying certain interpolation properties were classified. However, recently I have found another family of functions with similar properties. These new functions generalize Macdonald functions with t=0, have a combinatorial formula in terms of semi-standard tableaux and satisfy a version of the Cauchy identity. In my talk I will present these functions, describe their properties and will explain how these function were found using solvable lattice models and representations of quantum affine sl2 algebra. |

September 12th | Esme Bajo, UC Berkeley | ## Weighted Ehrhart TheoriesEhrhart theory—the study of lattice point enumeration in polytopes with rational vertices—can be used to study various combinatorial objects, including posets and graphs. In this talk, we explore two weighted versions of Ehrhart theory. We first ask which polynomial weights we can apply to our lattice so that the associated weighted h*-polynomials retain some of their classical properties, such as nonnegativity and monotonicity. We also study a second weighted Ehrhart theory, Chapoton’s q-analog Ehrhart theory, and discuss its relationship to the principal specialization of Stanley’s chromatic symmetric function. The first project is joint work with Robert Davis, Jesús A. De Loera, Alexey Garber, Sofía Garzón Mora, Katharina Jochemko, and Josephine Yu, and the second project is joint work with Matthias Beck and Andrés R. Vindas Meléndez. |

September 19th | Sam Armon, University of Southern California | ## Kohnert's rule for flagged Schur modulesTo any finite diagram $D$ is associated a $B$-module $S_D^{\mathrm{flag}}$ called the flagged Schur module. The characters of flagged Schur modules include Schur polynomials, key polynomials, and Schubert polynomials as special cases. We prove, for a family of diagrams $D$ called northwest diagrams, that the character of $S_D^{\mathrm{flag}}$ can be computed via a simple combinatorial rule called Kohnert's rule. Joint with Sami Assaf, Grant Bowling, and Henry Ehrhard. |

September 26th | Milo Bechtloff Weising, UC Davis | ## Stable-Limit Non-Symmetric Macdonald FunctionsNon-symmetric Macdonald polynomials play an important role in the representation theory of double affine Hecke algebras. These special polynomials give a basis for the standard DAHA representation consisting of weight vectors for the classical Cherednik operators and exhibit many interesting combinatorial properties related to affine Weyl groups. I will discuss a natural extension of these polynomials to the setting of the stable-limit DAHA of Ion-Wu. In this case we will obtain a basis for the standard stable-limit DAHA representation consisting of weight vectors for the limit Cherednik operators. These generally infinite variable functions exhibit combinatorial properties akin to their finite variable counterparts with some interesting differences. I will also discuss some further directions in this theory including links to the Shuffle Theorem of Carlsson-Mellit. |

October 3rd | Karola Mészáros, Cornell | ## Log-concavity of the Alexander polynomialThe central question of knot theory is that of distinguishing links up to isotopy. The first polynomial invariant of links devised to help answer this question was the Alexander polynomial (1928). Almost a century after its introduction, it still presents us with tantalizing questions, such as Fox’s conjecture (1962) that the absolute values of the coefficients of the Alexander polynomial $\Delta_L(t)$ of an alternating link $L$ are unimodal. Fox’s conjecture remains open in general with special cases settled by Hartley (1979) for two-bridged knots, by Murasugi (1985) for a family of alternating algebraic links, and by Ozsv\’ath and Szab\’o (2003) for the case of genus 2 alternating knots, among others. We settle Fox’s conjecture for special alternating links. We do so by proving that a certain multivariate generalization of the Alexander polynomial of special alternating links is Lorentzian. As a consequence, we obtain that the absolute values of the coefficients of $\Delta_L(t)$, where $L$ is a special alternating link, form a log-concave sequence with no internal zeros. In particular, they are unimodal. This talk is based on joint work with Elena Hafner and Alexander Vidinas. |

October 10th | Persi Diaconis, Stanford | ## Combinatorics and the Sylow theoremsSylow's theorem is basic group theory and the proof is 'just' easy combinatorics (nowadays). A standard proof involves understanding the Sylow-p subgroups of the symmetric group S_n. These are 'chandelier' groups and there is a nice description of their conjugacy ckasses and complex characters. BUT There are real mysteries about the Sylow double cosets in S_n: How many are there? What are their sizes? Are they 'nicely describable? Is the associated Hecke algebra 'nice' (e.g. Frobenius)? All these questions are open. I'll say what we know and try to say why anyone cares. I'll also try to explain it all 'in English'. |

October 17th | Sean Griffin, UC Davis | ## Partial resolutions of the nilpotent cone, and the Delta ConjectureThe Delta Conjecture gives two formulas for the result of applying a certain Macdonald eigenoperator to the elementary symmetric function e_n. One of these formulas, the "rise formula," has been proven independently by Blasiak, Haiman, Morse, Pun, and Seelinger, and by D'Adderio and Mellit. However, a Schur function expansion of this symmetric function is not known. In this talk, I will explain how the theory of partial resolutions of the nilpotent cone of Borho and MacPherson can be used to give a new Schur function expansion for the t=0 part of this symmetric function in terms of cocharge on "battery-powered tableaux". I will then explain how this formula can be extended to give conjectural Schur expansions for the t and t^2 coefficients in the Delta Conjecture symmetric function. Based on joint work with Maria Gillespie. |

October 24th | Daniel Orr, Virginia Tech | ## From quantum toroidal algebras to wreath Macdonald operatorsI will explain how quantum toroidal algebras (of type A) lead to the construction of a novel family of commuting difference operators having Haiman's wreath Macdonald polynomials as their joint eigenfunctions. Wreath Macdonald polynomials originate from the theory of symplectic quotient singularities, where the group involved in the quotient is a wreath product built from a symmetric group and an arbitrary finite cyclic group. When the cyclic group is trivial, one recovers the usual (type GL) Macdonald polynomials, whose theory is extensively developed. In contrast, very little is known about wreath Macdonald polynomials for higher order cyclic groups. Our 'wreath Macdonald operators' provide a new, direct characterization of these polynomials for arbitrary finite cyclic groups. |

October 31st | Mikhail Mazin, Kansas State | ## Recursions for rational Catalan series and homology of torus linksAnton Mellit and Matt Hogancamp introduced a recursion computing Khovanov-Rozansky homology of torus links. Together with Eugene Gorsky and Monica Vazirani, we showed that the generating series of (dn,dm)-invariant subsets with respect to the area and dinv statistics satisfy a recursion, equivalent to the Mellit-Hogancamp's, proving that these series (which we call rational (dn,dm) (q,t)-Catalan series) are equal to the Khovanov-Rozansky homology of the (dn,dm) torus link. In this talk I will introduce the dinv and area statistics on the invariant subsets, explain the recursion for the rational Catalan series, and show why it is equivalent to the Mellit-Hogancamp's recursion. Time permitting, I will also explain how one can modify the recursion so that only admissible invariant subsets are counted. As a result, the modified recursion computes the usual rectangular (dn,dm) (q,t)-Catalan polynomial, same as in the rational Shuffle Theorems. |

November 7th | Jonathan Novak, UC San Diego | ## How to Hurwitz and WhyHurwitz theory is a classical branch of enumerative geometry concerned with counting maps from curves to curves. It can be formulated entirely in terms of the combinatorics of permutations, and this elementary perspective has the advantage of revealing meaningful generalizations. I will explain one such generalization, monotone Hurwitz theory, which over the last decade has come to play an important role in random matrix theory and the asymptotic analysis of various special functions, specifically hypergeometric functions of matrix argument and partition functions of two-dimensional Yang-Mills theory. In addition to these analytic applications, it turns out that monotone Hurwitz theory also gives a very pretty way to understand the absolute order on the symmetric group, interacting beautifully with poset-theoretic notions such as R-labelings and lexicographic shellability. |

Thursday November 9th, 11:40am-1:00pm, Evans 1015 |
Patricia Hersh, Oregon | ## A relaxation of the notion of recursive atom ordering that still implies CL-shellabilityWhen Bjoerner and Wachs introduced one of the main forms of lexicographic shellability, namely CL-shellability, they also introduced the notion of recursive atom ordering, and they proved that a finite bounded poset is CL-shellable if and only if it admits a recursive atom ordering. We introduce a relaxation of the notion of recursive atom ordering, and we prove that any such relaxed recursive atom ordering may be transformed via a reordering process into a traditional recursive atom ordering. We use this to prove that several different notions of lexicographic shellability are all equivalent to each other, in the sense that any finite bounded poset admitting one of these admits all of them. As an application, we prove that the uncrossing orders, namely the face posets for stratified spaces of planar electrical networks, are dual CL-shellable. We will review background in topological combinatorics along the way and in particular will not assume familiarity with lexicographic shellability in this talk. This is joint work with Grace Stadnyk. |

November 14th | Colleen Robichaux , UCLA | ## Castelnuovo-Mumford regularity and excited Young diagramsWe introduce an algorithm which combinatorially computes the Castelnuovo-Mumford regularity of 321-avoiding Kazhdan-Lusztig varieties using excited diagrams. This extends previous work with Rajchgot and Weigandt which computes the regularity of Grassmannian Kazhdan-Lusztig varieties. We then discuss a specialization which computes the regularity of all two-sided mixed ladder determinantal varieties in terms of lattice paths. |

November 21st | No seminar - Thanksgiving | |

November 28th | Sylvie Corteel, UC Berkeley | ## Combinatorial formulas for Macdonald polynomials via interacting particle systemsWe describe some recently discovered connections between one-dimensional interacting particle systems and Macdonald polynomials and show the combinatorial objects that make this connection explicit. The first such model is the multispecies asymmetric simple exclusion process (ASEP) on a ring, linked to the symmetric Macdonald polynomial $P_{\lambda}(X;q,t)$ through its partition function, with multiline queues as the corresponding combinatorial object. The second particle model is one of the multispecies totally asymmetric zero range process (TAZRP) on a ring, which was recently found to have an analogous connection to the modified Macdonald polynomial $\widetilde{H}_{\lambda}(X;q,t)$. I will also discuss first steps towards a combinatorial formula for Koorwinder polynomials (Macdonald polynomials of type BC_n) using ASEP with open boundaries. This is joint work with David Keating, Olya Mandelshtam and Lauren Williams. |

December 5th | Shunsuke Tsuchioka, Tokyo Tech |