02 October, 2017

Fall 2017

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

29 May, 2017

Spring 2017

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

23 January, 2017

Hilbert Schemes and Macdonald Positivity

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)

25 May, 2016

Spring 2016

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

Kirill Paramonov presented a combinatorial approach to crystal base theory directly using the Stembridge axioms. This is based on an upcoming book on the topic by Dan Bump and Anne Schilling. He also explained how the crystal approach can be used to explain the expansion of the Stanley symmetric function into Schur functions.

Roger Tian presented his thesis work on the characterization of rigged configurations in B(infinity), which is part of his joint work with Hyenomi Lee and Jin Hong on the topic.

Henry Kvinge gave a talk on his work on Khovanov's Heisenberg category and shifted symmetric functions in joint work with Licata and Mitchell.

Federico Castillo gave an impressionistic view of Haiman's work on the n! and (n+1)^(n-1) theorems. He also prepared a nice handout.

27 January, 2016

Cyclotomic Hecke algebras.

Henry Kvinge gave a presentation on the representation theory of cyclotomic Hecke algebras (or representations of the symmetric group over a field of positive characteristic if you prefer). This was based on the expository paper 'Representations of symmetric groups and related Hecke algebras' by Alexander Kleshchev. Henry discussed how the symmetric group has special elements called Jucys-Murphy elements which commute. By studying the generalized eigenvalues of these elements on representations, he showed how we can break representations up into blocks.

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.

13 November, 2015

Strange expectations.

Kirill Paramonov gave a presentation on the paper by Marko Thiel and Nathan Williams (arXiv: 1508.05293). The authors base their work on methods by Paul Johnson who described the bijection between (n,m)- core partitions and A_n type lattice points in some simplex mP. Thiel and Williams generalize the notion of (n,m)-cores by looking at the colattices Q generated by simply-laced root systems. They introduce (W, b)- cores for any affine Weyl group W and calculate the generalized size statistic as a statistic on points of Q. Then, in the spirit of Johnson's paper, they use the results from Ehrhart theory to find the first and the second moment of the size statistic.

05 November, 2015

The Knizhnik-Zamolodchikov functor

Oscar Kivinen gave a presentation on the Knizhnik-Zamolodchikov (KZ) functor for the rational Cherednik algebra. The lecture was based on Chapter 6 of Etingof's second set of notes and notes by Gwyn Bellamy. The KZ functor sends modules in category O to Hecke algebra modules. We start with a representation inside category O of the rational Cherednik algebra of a complex reflection group G. After localization, this can be considered as a G-equivariant vector bundle with flat connection (or equivalently, a D-module) on the complement of the reflection hyperplanes in the reflection representation of G, using the Dunkl embedding. Since the action is free, this descends to a bundle on the quotient space h_{reg}/G. Taking horizontal sections of the corresponding connection (the KZ connection), we recover a local system that corresponds to a representation of the fundamental group of the base space, which is called the braid group. In the end, we sketched a proof that this representation actually comes from a representation of the cyclotomic Hecke algebra of G, or more precisely that the KZ functor factors through the category of H_q(\pi_1(h_{reg}/G))-modules. As an example, we computed the image of the KZ functor on Verma modules in type A_1. The KZ functor can be shown to be an equivalence of O/ O_{tor}, where O_{tor} is the torsion part of category O, namely the modules supported on the hyperplanes. In particular, the KZ functor can be (hoped to be) used to deduce useful information about Hecke algebra modules, such as computing Homs, and vice versa.

Rational Chendrik algebra

Gicheol Shin presented one chapter of the notes by Pavel Etingof and Xiaoguang Ma (arXiv:1001.0432v4), which covers the definition of rational Cherednik algebras and category O of a rational Cherednik algebra. As the universal enveloping algebra of a semisimple Lie algebra has the triangle decomposition, U(n-) x U(h) x U(n), a rational Cherednik algebra also has triangle decomposition, which follows from the PBW theorem for RCAs. Thus, we can define naturally the notion of category O for RCAs. As a category, the category O has very nice properties: it is abelian, closed under sub/quotient/extensions, has finite length, and a highest weight category relative to the poset Irrep(W), the isomorphic class of finite dimensional irreducible representation of W.

07 October, 2015

Lattice Points and Simultaneous Core Partitions.

Kirill Paramonov presented a paper by Paul Johnson (arXiv:1502.07934v2) that applies results from Ehrhart theory to the study of simultaneous (a,b)-core partitions. The basic idea is based on the bijection between the set of (a,b)-cores and integer points z = (z_1, z_2, ..., z_a) in b-th iteration of (a-1)-dimensional simplex, satisfying z_i >= 0 ; z_1 + z_2 + ... + z_a = b and 1*z_1 + 2*z_2 + ... + a*z_a = 0 (mod a). Equivalently, there's a bijection between (a,b)-cores and Z_a representations of degree b and with trivial determinant. Under this bijection, it's easy to find the total number of (a,b)-cores, which is known to be the rational Catalan number. For fixed a, the average size of cores turnes out to be a quadratic polynomial with respect to b. Therefore, in order to show that the average size of an (a,b)-core is (a+b+1)(a-1)(b-1)/24, we only need to verify this for three values of b. There are lots of other applications of this method. One can apply Ehrhart theory to get interesting results about q,t-generalization of Catalan numbers Cat(q,t) as function of a and b.

30 September, 2015

Random to top shuffle operators

Roger Tian gave a practice talk for his AMS talk in Chicago October 2-3 on his work on top to random shuffling, in which the first a cards are removed from a deck of n cards 12...n and then inserted back into the deck. He gave an expansion formula for k iterations of the top to random shuffle, generalizing a previous formula of Adriano Garsia. Then he further generalized the expansion formula to the situation of top to random shuffling a deck whose cards have multiple faces. These expansion formulae can be used for enumeration and calculating probabilities.

16 September, 2015

Eugene Gorsky and Anne Schilling discussed some reading material we would like to go over in the coming month for the informal seminar.

1. Representation theory:
• 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

2. Combinatorics:
• Armstrong, Loehr, Warrington on rational Catalan combinatorics: pdf
• Bergeron et al on rational shuffle conjecture: pdf
• Connections to Erhart theory: Johnson pdf, Thiel-Williams pdf
• Carlsson-Mellit proving shuffle conjecture: pdf

05 September, 2015

q,t-Catalan polynomials

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!

10 October, 2013

Analogue of a rim-hook rule for quantum Schubert calculus

Nate Gallup reported on his undergraduate thesis he wrote with Liz Beazley. He explained the rim-hook rule in the Grassmannian case. It is a rule to compute products in the quantum Grassmannian from the classical product in terms of Littlewood-Richardson coefficients. He then outlined what is known in the case of flag varieties (namely the Monk rule) and showed that a rim hook rule analogous to the Grassmannian case cannot exist.

28 May, 2013

Virtual crystals

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.

Splitting the square of a Schur function

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.

16 December, 2012

Markov chains on hyperplane arrangements

Travis Scrimshaw continued his presentation on multiple Dirichlet series. He showed how the Weyl group multiple Dirichlet series can be constructed from Gelfand Tsetlin patterns (at least in type A). He also clarified some of the issues during his last presentation regarding the multiplicative structure of the Dirichlet characters.

Federico Castillo started his presentation on random walks on hyperplane arrangements. He followed papers by Brown and Brown, Diaconis. In particular, we focused on braid arrangements.

05 December, 2012

Weyl group multiple Dirichlet series

Travis Scrimshaw gave a first introduction to Weyl group multiple Dirichlet series. He followed notes by Billey and her student in preparation for the ICERM program "Automorphic Forms, Combinatorial Representation Theory and Multiple Dirichlet Series". See also additional reading material here.

Roger Tian gave a new version of his qualifying exam talk. It was more focused on shuffle algebras and methods on how to find eigenvalues of shuffle operators.

19 November, 2012

Card shuffling

Roger Tian gave a first version of his qualifying exam talk. He started by presenting card shuffling and results of Diaconis and Garsia regarding certain identities for card shuffling operators. He then mentioned that he has a combinatorial proof for one of them. He proceed by stating open problems regarding the Murnaghan-Nakayama rule for k-Schur functions. Given all our comments, he will prepare a revised and more concise version for next week.

15 November, 2012

Weak horizontal strips

Mark Lydon continued his presentation on the paper by Krob and Thibon on Noncommutative symmetric functions. IV. Quantum linear groups and Hecke algebras at q=0." J. Algebraic Combin. 6 (1997), no. 4, 339-376 on Friday.

On Wednesday Leila Kadir presented the proof in the paper by Lapointe and Morse "Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions" arXiv:0402.530 that weak horizontal strips can be characterized via the action of the affine symmetric group on cores or, equivalently, as horizontal strips and vertical strips under their k-conjugation.

28 October, 2012

Representation theory of the 0-Hecke algebra

Mark Lydon started his presentation on the paper by Krob and Thibon on Noncommutative symmetric functions. IV. Quantum linear groups and Hecke algebras at q=0." J. Algebraic Combin. 6 (1997), no. 4, 339-376. In this first meeting he focused on the representation theory of the 0-Hecke algebra and associated combinatorics.

Supercharacters

Alex Lang talked to us about his reading and ideas on supercharacter theory.

16 October, 2012

Solomon's descent algebra

Roger Tian presented the paper by A. Garsia and C. Reutenauer "A decomposition of Solomon's descent algebra". We discussed the various bases presented in the paper.

03 October, 2012

Bijective proof shuffling theorem

Roger Tian presented his own bijective proof of Theorem 1.3 in Adriano Garsia's paper "On the powers of top to random shuffling". Together we worked out a more explicit formulation of the inverse bijection.

04 September, 2012

Shuffling algebras

Roger Tian presented the paper by Adriano Garsia "On the powers of top to random shuffling". This paper proves some results of Diaconis et al. on eigenspaces of the shuffling operators in purely algebraic and combinatorial terms. We worked out many examples by hand and are ready to program!

09 April, 2012

Markov chains

Arvind Ayyer, Steve Klee and Anne Schilling met many times this past quarter to discuss Markov chains defined by generalized promotion operators. Sch\"utzenberger defined a promotion operator on the set L of linear extensions of a finite poset of size n. This gives rise to a strongly connected graph on L. By assigning weights in two different ways to the edges of the graph, we study two Markov chains, both of which are ergodic. The stationary state of one gives rise to the uniform distribution, whereas the weights of the stationary state of the other has a nice product formula. This generalizes results by Hendricks on the Tsetlin library, which corresponds to the case when the poset is the anti-chain and hence L=S_n is the full symmetric group. We also provide the eigenvalues and eigenvectors of the transition matrix in general when the poset is a union of chains.

One-Skeleton Galleries

Travis Scrimshaw presented one-skeleton galleries which are used in describing the Gaussent-Littelmann formula for calculating Hall-Littlewood polynomials. One-skeleton galleries are collections of vertices and edges of the standard apartment of the affine building, and in particular all one needs are 2-step one-skeleton galleries to compute the Gaussent-Littelmann formula.

18 January, 2012

A combinatorial formula for fusion coefficients

Anne Schilling talked about joint work with Jennifer Morse on a combinatorial formula for fusion coefficients. Using the expansion of the inverse of the Kostka matrix in terms of tabloids as presented by Egecioglu and Remmel, she showed that the fusion coefficients can be expressed as an alternating sum over cylindric tableaux. Cylindric tableaux are skew tableaux with a certain cyclic symmetry. When the skew shape of the tableau has a cutting point, meaning that the cylindric skew shape is not connected, we give a positive combinatorial formula for the fusion coefficients. The proof uses a slight modification of a sign-reversing involution introduced by Remmel and Shimozono. It was also discussed how this approach may work in general.

11 January, 2012

Open problems of Artin groups

Travis Scrimshaw presented Artin groups, described some known properties and theorems, and discussed some open problems and conjectures.

19 November, 2011

Non-Commutative Schur Functions

Travis Scrimshaw presented a non-commutative version of Schur functions that were introduced by Fomin and Greene which are a generalization of the Platic Schur functions that Scrimshaw briefly discussed in his last talk. The discussed non-commutative Schur functions are in variables which satisfy non-local Knuth relations (or the stronger condition, non-local commutativity) and one local relation. These non-commutative Schur functions commute and satisfy all other expected Schur function relations (ex. Littlewood-Richardson rule).

07 November, 2011

Quasisymmetric functions

Jeff Ferreira presented the results of his dissertation. First he discussed a Littlewood-Richardson type rule for a basis of quasisymmetric functions called row-strict quasisymmetric Schur functions. This basis is defined as the generating functions for appropriately defined composition shaped tableaux. The Littlewood-Richardson type rule shows that the product of a row-strict quasisymmetric Schur function with a symmetric Schur function decomposes into a nonnegative integral sum of row-strict quasisymmetric Schur functions.

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.

01 November, 2011

The Plactic Monoid

Travis Scrimshaw presented the plactic monoid which encapsulates the combinatorics of Young tableaux. Consider the totally ordered alphabet A = [n], then denote by A^* the free monoid generated by A. The Robinson-Schensted-Knuth (RSK) algorithm takes any word w in A^* to a semi-standard Young tableau P and a standard Young tableau Q. Knuth described a congruence ~ on A^* such that if w ~ w' then P(w) = P(w'). The plactic monoid is defined as Pl(A) = A^* / ~, and using this, we proved Greene's invariants on w. Additionally from Z[Pl(A)], we defined Schur functions and demonstrated that they correspond to classical Schur functions.

25 October, 2011

Projective invariants of vector configurations

Andrew Berget presented his current work with Alex Fink on projective invariants of vector configurations. We have a left action of GL_r on r by n matrices and a right action by the torus. The projective equivalence class of v=(v_1,...,v_n) is the orbit GL_r v T. Andy presented a theorem by himself and Alex Fink which gives a necessary and sufficient condition for u to be in the closure of GL_r v T by linear dependence relations between u_{i_j} \otimes (v_J^\dual)_j, where dual here is Gale duality and J a subset of [n]. They conjecture that the ideal generated by the list of polynomials produced by this is the same as the prime ideal of the closure of GL_r v T. In addition, the multigraded Hilbert series of the closure of GL_r v T gives rise to the Tutte polynomial.

22 October, 2011

Extended Promotion on Posets

Arvind Ayyer presented conjectures on random walks on graphs defined by an extensions of promotion on linear extensions of posets. Let P be a poset of size n and L a linear etension of P. The elements in L can be viewed as permutations. Stanley defined the action of promotion in this setting. This action can be generalized by only doing promotion moves for elements larger than m for 1<=m<=n. This action imposes a graph structure on L and Arvind has a conjecture for the distribution of random walks on this graph with various ways of assigning weights.

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.

29 April, 2011

Charge and Energy on Affine Crystals

Prof. Anne Schilling presented recent work with Christian Lenart on the charge statsitic and its relation to energy functions on affine crystals. In particular, she presented the charge function on tableau and demonstrated how it can be related to the energy function on a tensor product of crystals in Type A. Prof. Schilling also discussed strategies to extend the charge statistic to other types in order to obtain a statistic consistent with affine energy functions.

20 April, 2011

Cohomology of Torus Orbits in Grassmannians.

Andy Berget described several approaches to understanding the coholomogy class of a torus orbit closure in a Grassmannian G_r(n). The approaches were:

(1) Using intersection theory to do the computation in H*(G_2(n)).

(2) Use GKM to do a computation in H*_T(G_r(n)), computing the class in the equivariant cohomology, and producing a very intriguing formula.

(3) Look at K-classes of orbit closures of matrices.

His work on this is joint with Alex Fink.

03 April, 2011

While Prof. Schilling is in Germany, we have coordinated a reading group for those interested in crystal bases and crystal graphs for their research. Regular participants include Tom Denton, Jeff Ferreira, Yvonne Kemper, Alex Lang, Katie O'Reilly, Travis Scrimshaw, and Rabhar Virk (rv).

Week 1: Tom Denton led an organizational meeting, and reprised an introduction to combinatorial crystal graphs that he gave in Kenya in December.

Week 2: Katie O'Reilly continued the introduction to combinatorial crystals, describing the tensor product rule and working through examples with the group.

Week 3-4: Jeff Ferreira reviewed the representation theory of gl_n, and gave an overview of crystals arising from Demazure modules and their relation to classical highest weight crystals.

Week 5: Tom Denton discussed the Stembridge local characterization of crystal bases for simply laced types.

Week 6-8: Travis Scrimshaw and Tom Denton presented the crystal structure on rigged configurations, including the bijection between rigged configuration and crystal elements, the algorithm for determining the crystal operators, and a few worked examples.

With one missed week and an organizational meeting, this filled out the whole term.

01 December, 2010

Cyclage Trees

Katie O'Reilly continued her previous talk by discussing the cyclage operator on words and the construction of the resulting cyclage trees on the corresponding tableaux. She explained how certain statistics of these trees relate to crystal graphs, and gave another description of the Kostka-Foulkes polynomial in terms of the charge statistic, which can be read from a cyclage tree.

24 November, 2010

The Plactic Monoid and Crystal Graphs

Katie O'Reilly presented an introduction to crystal graphs, based on the chapter "The Plactic Monoid" by Alain Lascoux and the paper "Crystal Graphs and q-Analogues of Weight Multiplicities for the Root System A_n" by Lascoux, Leclerc, and Thibon. She began by reviewing Young Tableaux and the Schensted Algorithm for inserting words into tableaux, then defined the Plactic Monoid by way of the Knuth Relations (which determine when two words produce the same tableaux under Schensted insertion). From here, the coplactic operators e_i, f_i, and sigma_i were introduced and used to construct the crystal graph associated to a tableau shape. Katie defined several statistics on the crystal graph and used these to provide an alternate description of the Kostka-Foulkes polynomial.

05 November, 2010

k-Schur Functions and Tilting Modules

Prof. Anne Schilling gave an introduction to k-Schur functions with the intent of building a connection to tilting modules. She first reviewed a number of improtant results in symmetric function theory for the purpose of generalization to the k-Schur case, in particular the Pieri rule and the connection between symmetric functions and the cohomology of Grasssmannians. The k-Schurs were then defined, with a presentation of the bijection between k-bounded partitions and k+1-cores, which led to a discussion of the k-Pieri rule, for which a geometric description is still desired. Finally, some connections between the k-Littlewood-Richardson rule and some other important structure constants (such as the fusion coefficients and structure constants for quantum cohomology) were mentioned.

12 October, 2010

Category O and Tilting Modules

Rahbar Virk (known to most as Rv) gave two presentations, in the Student Run and regular Discrete seminars; these presentations took the place of our regular meeting.

Rv's first presentation was a very accessible introduction to the finite dimensional representation theory of Lie algebras of finite type, which then led to the construction of Verma modules as infinite dimensional irreducible modules for these algebras. Rv then gave a definition of the Grothendieck group for the category of representations of a finite dimensional Lie algebra, as well as the definition of the Category O of representations that are finitely generated, h-semi-simple, [b,b]-locally nilpotent (with respect to the Borel subalgebra).

In the Discrete seminar, rv introduced a number of modules in the principle block of category O which provide different bases for the Grothendieck group. He then discussed the problem of enumerating decomposition numbers of these modules in terms of one another, and the relation of these decomposition numbers to Kazhdan-Lusztig theory. He also described a proof of one of these decomposition numbers given in his thesis.