- The Galois Group
- Activities
- The Graduate Program
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- Student-run Seminars
- Technical Tutorials
- About Davis
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- The Galois Group
- Activities
- The Graduate Program
- Funding
- Other Advice
- Student-run Seminars
- Technical Tutorials
- About Davis
- External Links

srdisc

Information can be found at Math department courses webpage and the UC Davis Math Dept's Seminar Page.

Please join us for the Student Discrete Math Seminar. Although we don't have pizza, the talks are always interesting. Talk to the organizers if you have any questions or would like to speak at the seminar.

The most commonly asked question, especially for new graduate students, is: "I would
like to speak, but I don't know what to talk about! What can I give a talk on?"
Below we have some suggestions. Disclaimers: This list does not in any way attempt to cover
all of Discrete Mathematics, but instead focuses on research areas that are more
related to UC Davis specialties. In fact, the list does not even exhaust all of the topics
that UC Davis *does* specialize in, but only what we happened to think of at the time.
You are encouraged to seek out other topics and papers. Although there has been an attempt
to organize and categorize, many papers and books belong in several categories, so take it
all with several grains of salt.

The list is a work in progress - check back often for updates.

The following is an old listing of topics. We'll leave it up until we finish the new, more organized list.

- the Edelmann-Greene paper on what we now call Edelmann-Greene insertion (actually maybe some of what i'm (Alex Woo) thinking about is actually in the Billey-Jockhush-Stanley paper),
- the Billey-Warrington paper on 321-hexagon-avoiding permutations. Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations; http://arxiv.org/abs/math.CO/0005052
- There is an extensive list of papers at the lower part of the webpage http://www.math.ucdavis.edu/~vazirani/S05/KL.details.html
- other interesting papers might be Sarah Mason's recent work on a different RSK type algorithm related to nonsymmetric Schur functions. Or some papers on Key polynomials.
- Kevin Purbhoo's paper on root games
- Almost any section of Macdonald's book. You might also look at
Fulton's book or Jim Haglund's new book. There are several papers on
crystal graphs I can also suggest (for instance, in connection to the
Littlewood-Richardson rule).

Below are listed several papers related to the saturation conjecture/theorem. (In a slightly weird format from cutting/pasting from a bibtex file.) In particular there are several papers involving polytopes, honeycombs, Littelmann paths, galleries, and buildings.

- Arkady Berenstein, Andrei Zelevinsky "Tensor product multiplicities, canonical bases and totally positive varieties"
- math.RT/9912012
( Invent. Math. 143 (2001), no. 1,
77--128.)

[We obtain a family of explicit "polyhedral" combinatorial expressions for multiplicities in the tensor product of two simple finite-dimensional modules over a complex semisimple Lie algebra. Our answers use a new combinatorial concept of $\ii$-trails which resemble Littelmann's paths but seem to be more tractable. A remarkable observation by G. Lusztig notes that combinatorics of the canonical basis is closely related to geometry of the totally positive varieties. We formulate this relationship in terms of two mutually inverse transformations: "tropicalization" and "geometric lifting."] - Anders S. Buch "The
saturation conjecture (after A. Knutson and T. Tao)"

math.CO/9810180 (With an appendix by William Fulton. Enseign. Math. (2) 46 (2000), no. 1-2, 43--60. ) [A nice exposition of the hive model and Knutson-Tao's proof of the saturation conjecture (in type A).] - Joel Kamnitzer "Mirkovic-Vilonen
cycles and polytopes"

math.AG/0501365 [We give an explicit description of the Mirkovic-Vilonen cycles on the affine Grassmannian for arbitrary complex reductive groups. We also give a combinatorial characterization of the MV polytopes. We prove that a polytope is an MV polytope if and only if it a lattice polytope whose defining

hyperplanes are parallel to those of the Weyl polytopes and whose 2-faces are rank 2 MV polytopes. As an application, we give a bijection between Lusztig's canonical basis and the set of MV polytopes.] - M. Kapovich. "Generalized
triangle inequalities and their applications",
- Madrid, August 22-30, 2006. Eds. Marta Sanz-Solé, Javier Soria,
Juan L. Varona, Joan Verdera. Vol. 2, p. 719-742.
[survey article.
applications to decomposing tensor products of irreducible
representations and the saturation theorem in types beyond A.]

- Allen Knutson, Terence
Tao "The
honeycomb model of GL(n) tensor products I: proof of the saturation
conjecture"
math.RT/9807160
( J. Amer. Math. Soc. 12 (1999), no. 4, 1055--1090. )

[We introduce the honeycomb model of BZ polytopes, which calculate Littlewood-Richardson coefficients, the tensor product rule for GL(n). A particularly well-behaved honeycomb is necessarily integral, which proves the "saturation conjecture", extending results of Klyachko to give a complete answer to which L-R coefficients are positive. This in turn has as a consequence Horn's conjecture from 1962 characterizing the spectrum of the sum of two Hermitian matrices. ] - Sophie Morier-Genoud's thesis: "Relevement Geometrique de L'involution de Schutzenberger et Applications" [in French]?
- Arun Ram "Alcove
walks, Hecke algebras, spherical functions, crystals and column strict
tableaux"
math.RT/0601343
(Pure and Applied Mathematics Quarterly. Special Issue: In honor of
Robert MacPherson. Vol 2. (2006) 135-183.)
[This paper makes precise
the close connection between the affine
Hecke algebra, the path model, and the theory of crystals. Section 2 is
a
basic pictorial exposition of Weyl groups and affine Weyl groups and
Section 5
is an exposition of the theory of (a) symmetric functions, (b) crystals
and
(c) the path model. Sections 3 and 4 give an exposition of the affine
Hecke
algebra and recent results regarding the combinatorics of spherical
functions
on p-adic groups (Hall-Littlewood polynomials). The $q$-analogue of the
theory
of crystals developed in Section 4 specializes to the path model
version of
the ``classical'' crystal theory. The connection to the affine Hecke
algebra
and the approach to spherical functions for a $p$-adic group in
Nelsen-Ram was
made concrete by C. Schwer who told me that ``the periodic Hecke module

encodes the positively folded galleries'' of Gaussent-Littelmann. This paper is a further development of this point of view.] - Guy Rousseau : "Euclidean
buildings"
Lecture notes from Summer School 2004: Non-positively curved
geometries, discrete groups and rigidities
http://www-fourier.ujf-grenoble.fr/ECOLETE/ecole2004/Rousseau.pdf

[Introduction to buildings. All basic definitions can be found here.] - Arkady Berenstein, David Kazhdan

Lecture notes on Geometric Crystals and their combinatorial analogues

math.QA/0610567

- A path model for geodesics in Euclidean buildings and its applications to representation theory, Michael Kapovich and John J. Millson, arXiv:math.RT/0411182
- {Gaussent, S. and Littelmann, P.}, {L{S} galleries, the path model, and {MV} cycles}, {Duke Math. J.}, {Duke Mathematical Journal}, VOLUME {127}, YEAR {2005}, PAGES {35--88},
- {Littelmann, Peter}, {The path model for representations of symmetrizable {K}ac-{M}oody algebras}, BOOKTITLE = {Proceedings of the International Congress of Mathematicians, Vol.\ 1, 2 (Z\"urich, 1994)}, PAGES = {298--308}, PUBLISHER = {Birkh\"auser}, YEAR = {1995},
- {Littelmann, Peter}, {Paths and root operators in representation theory}, {Ann. of Math. (2)}, VOLUME = {142}, YEAR = {1995}, PAGES = {499--525},
- {Berenstein, Arkady and Zelevinsky, Andrei}, {Canonical bases for the quantum group of type {$A\sb r$} and piecewise-linear combinatorics}, {Duke Math. J.}, VOLUME = {82}, YEAR = {1996}, PAGES = {473--502},
- {Berenstein, A. D. and Zelevinsky, A. V.}, {Tensor product multiplicities and convex polytopes in partition space}, {J. Geom. Phys.}, VOLUME = {5}, YEAR = {1988}, PAGES = {453--472},
- Galleries, Hall-Littlewood polynomials and structure constants of the spherical Hecke algebra, {Christoph Schwer}, {arXiv:math.CO/0506287}
- {De Loera, Jes{\'u}s A. and McAllister, Tyrrell B.}, {Vertices of {G}elfand-{T}setlin polytopes}, {Discrete Comput. Geom.}, VOLUME = {32}, YEAR = {2004}, PAGES = {459--470},
- Knutson, Allen and Tao, Terence and Woodward, Christopher, A positive proof of the {L}ittlewood-{R}ichardson rule using the octahedron recurrence, {Electron. J. Combin.}, VOLUME = {11}, YEAR = {2004},
- Knutson, Allen and Tao, Terence, {The honeycomb model of {${\rm GL}\sb n({\bf C})$} tensor products. {I}. {P}roof of the saturation conjecture}, {J. Amer. Math. Soc.}, VOLUME = {12}, YEAR = {1999}, PAGES = {1055--1090},
- Knutson, Allen and Tao, Terence and Woodward, Christopher, {The honeycomb model of {${\rm GL}\sb n(\mathbb C)$} tensor products. {II}. {P}uzzles determine facets of the {L}ittlewood-{R}ichardson cone}, {J. Amer. Math. Soc.}, VOLUME = {17}, YEAR = {2004}, PAGES = {19--48 (electronic)},
- Knutson, Allen and Tao, Terence , {Honeycombs and sums of {H}ermitian matrices}, {Notices Amer. Math. Soc.}, VOLUME = {48}, YEAR = {2001}, PAGES = {175--186},
- Buch, Anders Skovsted, {The saturation conjecture (after {A}.\ {K}nutson and {T}.\ {T}ao)}, {With an appendix by William Fulton}, {Enseign. Math. (2)}, VOLUME = {46}, YEAR = {2000}, PAGES = {43--60},
- Regular triangulations and the secondary polytope - a good source is chapter 7 of the Gelfand-Kapranoz-Zelevinsky book

Further suggestions are welcome!

srdisc.txt · Last modified: 2016/02/16 09:29 by jasnyder