### Fall 2023

During Fall 2023, David Kenepp, Mary Claire Simone, Regina Zhou, Lisa Johnston, Evuilynn Nguyen, Maria Mihaila, and Anne Schilling ran a weekly reading group on crystal graphs and applications.

A Web Log of ongoing discussions on Algebraic Combinatorics.

During Fall 2023, David Kenepp, Mary Claire Simone, Regina Zhou, Lisa Johnston, Evuilynn Nguyen, Maria Mihaila, and Anne Schilling ran a weekly reading group on crystal graphs and applications.

During Spring 2023, David Kenepp, Joseph Pappe, Mary Claire Simone, Regina Zhou,Lisa Johnston, Evuilynn Nguyen, and Anne Schilling ran a weekly reading group on representations of the symmetric group.

During Fall 2022, Jon Ericksson, David Kenepp, Joseph Pappe, Mary Claire Simone, Regina Zhou, and Anne Schilling ran a weekly reading group on Thrall's problem.

Here is a link to our seminar for the Winter Quarter 2021.

Here is a link to our seminar for the Fall Quarter 2020.

During the Spring Quarter 2018, we meet Mondays 2-3pm. We will focus on geometric crystals.

April 16 **Wencin Poh** Introduction to crystals

April 23 **Jose Simental Rodriguez** Perfect bases, crystals and functions on U

April 30 **Maria Gillespie** Geometric pre-crystals and crystals (Section 2 of Berenstein and Kazhdan I)

May 7 **Jianping Pan** Tropicalization

May 14 **Gicheol Shin** PhD Thesis: The Rectangular representation of the rational Cherednik algebra of type A

May 21 **Wencin Poh** An example of tropicalization (based on Lam Whittaker functions, geometric crystals, and quantum Schubert calculus, arXiv:1308.5451)

During the Winter Quarter 2018, we meet Tuesdays 1-2pm. We will focus on symplectic resolutions.

February 13 **Jose** Moment maps and GIT

February 20 **Lang** Quiver varieties

February 27 **Oscar** Quantization

March 6 **Tudor Dimofte** Duality

March 13 ** Alex Barverman**

Some reference:

Losev's lectures on quantized quiver varieties: click here

Braden-Proudfoot-Licata-Webster: arXiv:1208.3863 and arXiv.1407.0964

Okounkov's lectures: arXiv.1701.00713

During the Fall Quarter 2017, we meet Mondays at 10am. We will mostly focus on Jian-Yi Shi, Lecture Notes in Mathematics, The Kazhdan-Lusztig Cells in Certain Affine Weyl Groups [1 ed.]. Here is the first tentative list of speakers:

October 2 **Monica Vazirani**: Overview

October 9 **Lang Mou** Introduction to Kazhdan-Lusztig polynomials

October 16 **Wencin Poh** Kazhdan-Lusztig cells

October 23 **Yue Zhao** Star operation

October 30 **Gicheol Shin** More on the star operation

November 6 **Mikhail Mazin** guest lecture

November 13 **Oscar Kivinen**

November 20 **Graham Hawkes**

November 27 **Eugene Gorsky**

December 4 currently open

During the Spring Quarter 2017, we have had the following presentations:

April 24 **Maria Gillespie**: LLT positivity, part I (after Haiman-Grojnowski)

May 1 **Anne Schilling** A Demazure crystal construction for Schubert polynomials

May 8 **Graham Hawkes** Crystal analysis of type C Stanley symmetric functions

May 15 **Oscar Kivinen** LLT positivity, part II

May 22 **Eugene Gorsky** q,t-Catalans and knots (after M. Hogancamp)

May 29 Memorial day

June 5 **Kirill Paramonov** Cores with distinct parts and bigraded Fibonacci numbers

This quarter, Maria and Oscar are going to organize an informal seminar on Hilbert schemes
and Macdonald positivity.

The two references for the seminar will be:

1. "Hilbert Schemes, Polygraphs, and Macdonald Positivity," https://math.berkeley.edu/~mhaiman/ftp/nfact/polygraph-jams.pdf

2. "Notes on Macdonald polynomials and the geometry of Hilbert schemes," https://math.berkeley.edu/~mhaiman/ftp/newt-sf-2001/newt.pdf

The first five weeks' topics can go something like this, to build up the background:

Week 1: Background, overview (Chapter 1 from Notes)

Week 2: Geometry of Hilbert Schemes (basic ideals perspective)

Week 3: The n! conjecture (Chapter 3 from Notes)

Week 4: Nested Hilbert Schemes and how Polygraphs come into play (Chapters 3.4, 4.1 from Notes)

Week 5: Polygraph Theorem Lecture 1 (starting from Hilbert schemes paper)

We have had several excellent presentations during the Springer Quarter 2016.

For every Young diagram we can construct a corresponding representation for a symmetric group called a Specht module. Over a field of characteristic zero these representations are simple but once we move to a field of characteristic p > 0, they may no longer be. In this case, the full set of simple representations appear as certain quotients of the Specht modules corresponding to p-regular (or p-restricted depending on your conventions) diagrams.

This is all relevant because much of the representation theory of rational Cherednik algebras can be understood through the representation theory of cyclotomic Hecke algebras via the KZ functor.

- Etingof's lectures #1 (lectures 6, 7, 11 and maybe 10): pdf
- Etingof's lectures #2 (especially sections 1-3, 5-6): pdf
- Etingof&Losev's lectures: pdf
- Gwyn Bellamy's lectures: html
- Leclerc-Thibon on canonical bases: pdf
- More advanced Losev's lectures: pdf
- Braden, Proudfoot, Licata and Webster on general framework of symplectic resolutions: pdf
- relation between rational Cherednik algebra to diagonal harmonics and q,t-Catalans, Gordon pdf

This past year we had lots of informal seminars on q,t-Catalan polynomials, the shuffle conjecture (now a theorem), card shuffling, random walks and rigged configurations. **Graham Hawkes**, **Kirill Paramanov**, **Roger Tian**, **Hyeonmi Lee**, **Travis Scrimshaw**, **Ryan Reynolds**, **Gwen McKinley**, and **Eric Slivken** presented topics!

Last week **Travis Scrimshaw** started his presentation about virtual crystals. He presented the main definition and will continue this week with the proof of alignments of virtual crystals.

In the past two weeks **Roger Tian** gave several talks on the paper:

MR1331743 (97b:05165) Reviewed
Carre, Christophe(F-ROUEN-I); Leclerc, Bernard(F-PARIS7-LI)
Splitting the square of a Schur function into its symmetric and antisymmetric parts. (English summary)
J. Algebraic Combin. 4 (1995), no. 3, 201-231.

This might lead to exciting connections to crystals, plethysms, and symmetric chain decompositions:

MR3010696 Pending
Hersh, Patricia(1-NCS); Schilling, Anne(1-CAD)
Symmetric chain decomposition for cyclic quotients of Boolean algebras and relation to cyclic crystals. (English summary)
Int. Math. Res. Not. IMRN 2013, no. 2, 463-473.

On Wednesday

Next, Jeff defined a family of polynomials called Demazure atoms. These polynomials can be defined in a number of ways, including a definition via divided difference operators and a definition as specialized nonsymmetric Macdonald polynomials. Jeff presented two new characterizations of Demazure atoms. The first as Gelfand-Tsetlin type patters (here the first row of the triangular array is a weak composition, and the inequalities of the array are slightly more intricate), and the second in terms of Lakshmibai-Sheshadri paths.

Finally, Jeff discussed a generalization of the combinatorial formula for nonsymmetric Macdonald polynomials. This generalization allows the "basement" of the composition diagram filling to be an arbitrary permutation. Jeff shows that the resulting polynomials are the simultaneous eigenfunctions of a family of commuting operators in the double affine Hecke algebra. This result is analogous to how nonsymmetric Macdonald polynomials appear as eigenfunctions.

We discussed a proof of a special case given in the paper by Hendricks "The stationary distribution of an interesting Markov chain", J. Appl. Prob. 9 (1972) 231-233.