Department of

Mathematics


Seminar Calendar
for Geometry and Combinatorics events the year of Friday, January 1, 2021.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, January 7, 2021

3:00 pm in Zoom,Thursday, January 7, 2021

On the shifted Littlewood-Richardson coefficients and Littlewood-Richardson coefficients

Khanh Nguyen Duc   [email] (Université de Lyon)

Abstract: We give a new interpretation of the shifted Littlewood-Richardson coefficients $f_{\lambda\mu}^\nu$ ($\lambda,\mu,\nu$ are strict partitions). The coefficients $g_{\lambda\mu}$ which appear in the decomposition of Schur $Q$-function $Q_\lambda$ into the sum of Schur functions $Q_\lambda = 2^{l(\lambda)}\sum\limits_{\mu}g_{\lambda\mu}s_\mu$ can be considered as a special case of $f_{\lambda\mu}^\nu$ (here $\lambda$ is a strict partition of length $l(\lambda)$). We also give another description for $g_{\lambda\mu}$ as the cardinal of a subset of a set that counts Littlewood-Richardson coefficients $c_{\mu^t\mu}^{\tilde{\lambda}}$. This new point of view allows us to establish connections between $g_{\lambda\mu}$ and $c_{\mu^t \mu}^{\tilde{\lambda}}$. More precisely, we prove that $g_{\lambda\mu}=g_{\lambda\mu^t}$, and $g_{\lambda\mu} \leq c_{\mu^t\mu}^{\tilde{\lambda}}$. We conjecture that $g_{\lambda\mu}^2 \leq c^{\tilde{\lambda}}_{\mu^t\mu}$ and formulate some conjectures on our combinatorial models which would imply this inequality if it is valid. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.

Thursday, January 14, 2021

3:00 pm in Zoom,Thursday, January 14, 2021

A crystal for stable Grothendieck polynomials

Jianping Pan   [email] (UC Davis)

Abstract: We construct a type A crystal, which we call the $\star$-crystal, whose character is the stable Grothendieck polynomials for fully-commutative permutations. This crystal is a K-theoretic generalization of Morse-Schilling crystal on decreasing factorizations. Using the residue map, we showed that this crystal intertwines with the crystal on set-valued tableaux given by Monical, Pechenik and Scrimshaw. We also proved that this crystal is isomorphic to that of pairs of semistandard Young tableaux using a newly defined insertion called the $\star$ -insertion. The insertion offers a combinatorial interpretation to the Schur positivity of the stable Grothendieck polynomials for fully-commutative permutations. Furthermore, the $\star$ -insertion has interesting properties in relation to row Hecke insertion and the uncrowding algorithm. This is joint work with Jennifer Morse and Anne Schilling. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.

Thursday, January 28, 2021

3:00 pm in Zoom,Thursday, January 28, 2021

On the Okounkov-Olshanski formula for standard tableaux of skew shapes

Alejandro H. Morales   [email] (University of Massachusetts Amherst)

Abstract: The classical hook length formula counts the number of standard tableaux of straight shapes. In 1996, Okounkov and Olshanski found a positive formula for the number of standard Young tableaux of a skew shape. We prove various properties of this formula, including three determinantal formulas for the number of nonzero terms, an equivalence between the Okounkov-Olshanski formula and another skew tableaux formula involving Knutson-Tao puzzles, and two q-analogues for reverse plane partitions, which complements work by Stanley and Chen for semistandard tableaux. We also give applications and several reformulations of the formula, including two in terms of the excited diagrams appearing in a more recent skew tableaux formula by Naruse. This is joint work with Daniel Zhu. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.

Thursday, February 4, 2021

4:00 pm in Zoom,Thursday, February 4, 2021

The 1/3-2/3 Conjecture for Coxeter groups

Yibo Gao   [email] (Massachusetts Institute of Technology )

Abstract: The 1/3-2/3 Conjecture, originally formulated in 1968, is one of the best-known open problems in the theory of posets, stating that the balance constant of any non-total order is at least 1/3. By reinterpreting balance constants of posets in terms of convex subsets of the symmetric group, we extend the study of balance constants to convex subsets C of any Coxeter group. Remarkably, we conjecture that the lower bound of 1/3 still applies in any finite Coxeter group, with new and interesting equality cases appearing. We generalize several of the main results towards the 1/3-2/3 Conjecture to this new setting: we prove our conjecture when C is a weak order interval below a fully commutative element in any acyclic Coxeter group (a generalization of the case of width-two posets), we give a uniform lower bound for balance constants in all finite Weyl groups using a new generalization of order polytopes to this context, and we introduce generalized semiorders for which we resolve the conjecture. We hope this new perspective may shed light on the proper level of generality in which to consider the 1/3-2/3 Conjecture, and therefore on which methods are likely to be successful in resolving it. This is joint work with Christian Gaetz. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.

Thursday, February 11, 2021

3:00 pm in Zoom,Thursday, February 11, 2021

Doing Schubert calculus with Bumpless Pipe Dreams

Daoji Huang   [email] (Brown University)

Abstract: Bumpless pipe dreams were introduced by Lam, Lee, and Shimozono in the context of back stable Schubert calculus. Like ordinary pipe dreams, they compute Schubert and double Schubert polynomials. In this talk, I will give a bijective proof of Monk's rule for Schubert polynomials, and show that the proof extends easily to the proof of Monk's rule for double Schubert polynomials. As an application, I will explain how to biject bumpless pipe dreams and ordinary pipe dreams using the transition and co-transition formulas, which are specializations of Monk's rule. If time permits I will also briefly discuss and share some work in progress on how bumpless pipe dreams can be used to compute products of certain Schubert polynomials in generalizations of the Littlewood-Richardson rule for Schur polynomials. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.

Thursday, February 18, 2021

3:00 pm in Zoom,Thursday, February 18, 2021

Determinantal ideals, pipe dreams, and non-intersecting lattice paths

Li Li   [email] (Oakland University)

Abstract: Determinantal ideals are ideals generated by minors of a variable matrix. They play an important role in both commutative algebra and algebraic geometry. A combinatorial approach to study a determinantal ideal is to apply Groebner degeneration to get a squarefree monomial ideal, and study the corresponding simplicial complex instead. Through this approach, some invariants such as Hilbert polynomials of the ideals can be expressed in terms of pipe dreams and non-intersecting lattice paths. In this talk, I will report recent work on double determinantal ideals, and in particular the combinatorial objects that play the role of non-intersecting lattice paths. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.

Tuesday, February 23, 2021

3:00 pm in Zoom,Tuesday, February 23, 2021

The Peterson Isomorphism and Quantum Cohomology of the Grassmannian

Elizabeth Milićević   [email] (Haverford College)

Abstract: The Peterson isomorphism directly relates the homology of the affine Grassmannian to the quantum cohomology of any finite flag variety. In the case of a partial flag, Peterson’s map is only a surjection, and one needs to quotient by a suitable ideal to map isomorphically onto the quantum cohomology. In this talk, we first provide an exposition of this parabolic Peterson isomorphism in the case of the Grassmannian. We then relate the Peterson isomorphism via Postnikov’s strange duality to several quantum-to-affine correspondences on the k-Schur functions representing the homology of the affine Grassmannian. This talk includes joint work with J. Cookmeyer, as well as L. Chen and J. Morse. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.

Thursday, February 25, 2021

3:00 pm in Zoom,Thursday, February 25, 2021

Equivariant quantum Pieri rule on cylindric shapes

Kaisa Taipale   [email] (University of Minnesota and C.H. Robinson)

Abstract: The equivariant quantum cohomology of Grassmannians is an amazing place to study the collision of a number of combinatorial and algebro-geometric phenomena. In this talk, I will introduce an equivariant quantum Pieri rule for Grassmannians (https://arxiv.org/pdf/2010.15395.pdf) that uses the machinery of cylindric shapes in novel ways: it is computationally elegant, requires no recursion, and puts in reach the solutions to a number of additional combinatorial problems. While emphasizing the natural geometric structure of the Grassmannian that this rule illuminates, I’ll also tell the story of this problem, which started with collaboration and conjecture at IAS and continued through periods of both dormancy and deep discussion, as well as the career changes of several of the collaborators. This is joint work with Elizabeth Milićević, Dorian Ehrlich, and Anna Bertiger. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.

Thursday, March 4, 2021

3:00 pm in Zoom,Thursday, March 4, 2021

Uncrowding Algorithm for Hook-Valued Tableaux

Wencin Poh (UC Davis)

Abstract: Whereas set-valued tableaux are the combinatorial objects associated to stable (Grassmann) Grothendieck polynomials, hook-valued tableaux are associated to stable canonical Grothendieck polynomials. We define a novel uncrowding algorithm for hook-valued tableaux. The algorithm “uncrowds” the entries in the arm of the hooks and yields a set-valued tableau and a column-flagged increasing tableau. This uncrowding algorithm intertwines with the crystal operators on hook-valued tableaux. We also provide an alternative uncrowding algorithm that “uncrowds” the entries in the leg instead of the arm of the hooks. As an application, we obtain various expansions of the canonical Grothendieck polynomials.

Thursday, March 11, 2021

3:00 pm in Zoom,Thursday, March 11, 2021

Newell-Littlewood numbers

Shiliang Gao   [email] (University of Illinois at Urbana-Champaign)

Abstract: The Newell-Littlewood numbers are defined in terms of their celebrated cousins, the Littlewood-Richardson coefficients. Both arise as tensor product multiplicities for a classical Lie group. They are the structure coefficients of the K. Koike-I. Terada basis of the ring of symmetric functions. Recent work of H. Hahn studies them, motivated by R. Langlands' beyond endoscopy proposal; we address her work with a simple characterization of detection of Weyl modules. This motivates further study of the combinatorics of the numbers. We consider analogues of ideas of J. De Loera-T. McAllister, H. Derksen-J. Weyman, S. Fomin-W. Fulton-C.-K. Li-Y.-T. Poon, W. Fulton, R. King-C. Tollu-F. Toumazet, M. Kleber, A. Klyachko, A. Knutson-T. Tao, T. Lam-A. Postnikov-P. Pylyavskyy, K. Mulmuley-H. Narayanan-M. Sohoni, H. Narayanan, A. Okounkov, J. Stembridge, and H. Weyl. This is joint work with Gidon Orelowitz and Alexander Yong. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.

Thursday, April 1, 2021

3:00 pm in Zoom,Thursday, April 1, 2021

Minimal elements in the limit weak order

Christian Gaetz   [email] (Massachusetts Institute of Technology)

Abstract: The limit weak order on an affine Weyl group was introduced by Lam and Pylyavskyy in their study of total positivity for loop groups. They showed that in the case of the affine symmetric group the minimal elements of this poset coincide with the infinite fully commutative reduced words and with infinite powers of Coxeter elements. We answer several open problems raised there by classifying minimal elements in all affine types and relating these elements to the classes of fully commutative and Coxeter elements. Interestingly, the infinite fully commutative elements correspond to the minuscule and cominuscule nodes of the Dynkin diagram, while the infinite Coxeter elements correspond to a single node, which we call the heavy node, in all affine types other than type A. No background on Weyl groups will be assumed. This talk is based on joint work with Yibo Gao. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.

Thursday, April 22, 2021

3:00 pm in Zoom,Thursday, April 22, 2021

Stability of Heisenberg Coefficients

Li Ying   [email] (University of Notre Dame)

Abstract: The Heisenberg product is an associative product defined on symmetric functions which interpolate between the ordinary product and the Kronecker product. Heisenberg coefficients are Schur structure constants of the Heisenberg product and generalization of both Littlewood-Richardson coefficients and Kronecker coefficients. In 1938, Murnaghan discovered that the Kronecker product of two Schur functions stabilizes. In 2014, Stembridge introduced the notion of stability for Kronecker triples which generalize Murnaghan's classical stability result. Sam and Snowden proved a conjecture of Stembridge characterizing stable Kronecker triples. I will show an analogous result for Heisenberg coefficients. I will also show how to construct Heisenberg stable triples using additive matrices. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.

Thursday, April 29, 2021

3:00 pm in Zoom,Thursday, April 29, 2021

The Unitary Dual Problem and Combinatorics

Wai Ling Yee   [email] (University of Windsor)

Abstract: One of the most important open problems in mathematics is the Unitary Dual Problem: given a group, classify its irreducible unitary representations. The most common approach to classifying unitary representations is to study representations admitting non-degenerate invariant Hermitian forms (these are classified), compute the signatures of the forms, and then determine when the forms are definite. Signature character formulas are very complicated due to the recursive nature of the formulas. However, it turns out that the complexity may be encoded by well-known combinatorial objects: signature character formulas involve signed Kazhdan-Lusztig polynomials (which turn out to be classical Kazhdan-Lusztig polynomials evaluated at $-q$ times a sign) and pieces of Hall-Littlewood polynomials (called Hall-Littlewood polynomial summands) evaluated at $q=-1$. Some of the material covered in this talk is joint work with Justin Lariviere. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.

Thursday, May 6, 2021

3:00 pm in Zoom,Thursday, May 6, 2021

Counting linear extensions of posets using determinants

Alexander Garver   [email] (Carleton College)

Abstract: We introduce a class of posets, which includes both ribbon posets (skew shapes) and d-complete posets, such that their number of linear extensions is given by a determinant of a matrix whose entries are products of hook lengths. If time permits, we will also present q-analogues of this determinantal formula in terms of the major index and inversion statistics. This is joint work with Stefan Grosser, Jacob Matherne, and Alejandro Morales. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.

Thursday, September 16, 2021

3:00 pm in 347 Altgeld Hall,Thursday, September 16, 2021

Grothendieck-to-Lascoux Expansions

Tianyi Yu   [email] (UCSD)

Abstract: We establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by permutations into Grassmannian stable Grothendieck polynomials. Our expansion is the K-theoretic analogue of that of a Schubert polynomial into Demazure characters, whose symmetric analogue is the expansion of a Stanley symmetric function into Schur functions. Our expansions extend to flagged Grothendieck polynomials. This is a joint work with Mark Shimozono.