My research interests range from algebraic combinatorics, representation theory to mathematical physics. I have been interested in combinatorial properties of quantum algebras. Quantum algebras have their roots in two dimensional solvable lattice models in statistical mechanics. Kashiwara showed that at zero temperature the quantum algebras exhibit beautiful combinatorial properties. Mathematically these are formulated in the crystal base theory. Particularly nice crystals are Kirillov-Reshetikhin crystals; they are affine finite-dimensional crystals that can be used in the Kyoto path model to construct highest weight affine crystals. They are also in close relation to Demazure crystals as shown in work with Fourier and Shimozono. Together with Okado and Fourier, I recently showed that all Kirillov-Reshetikhin crystals of nonexceptional types exist and constructed them combinatorially exploiting certain automorphisms of the affine Dynkin diagrams. With Kirillov and Shimozono I have established a relation between crystal base theory and other combinatorial objects, called rigged configurations, which also arise from physics. A generalization of this result provides new fermionic formulas of the physical partition sums which, as follows by work of the Stony Brook group, encapture the statistics of the particle in the physics model (actually these statistics are similar to those found in the fractional quantum Hall effect; see here). In terms of q-series these yield generalizations of the famous Rogers-Ramanujan identities.
I am also part of the Focused Research Group on Affine Schubert Calculus: Combinatorial, geometric, physical, and computational aspects funded by the NSF. More information can be found at the FRG website.
I am especially fond of understanding relations between
different areas in mathematics and mathematical physics.
Often this helps to unravel hard problems in one area by
exploiting the techniques of another area. I have also contributed to
the open-source computer package Sage
through Sage-Combinat
by implementing crystal bases and Kirillov-Reshetikhin crystals
(I previously contributed to MuPAD-Combinat).
If you want to learn more on how to use crystals in Sage, there is a thematic tutorial on
Lie methods and related combinatorics
written in collaboration with Dan Bump from Stanford.
(Parabolic) quantum Bruhat graphs for A2 and A3 with J={1}
Some of my papers
with Toufik Mansour
ECA 3:2 (2023) Interview #S3I5 ( https://doi.org/10.54550/ECA2023V3S2I5 )
with Joseph Pappe, Stephan Pfannerer, Mary Claire Simone
preprint arXiv:2212.13588
with Laura Colmenarejo, Rosa Orellana, Franco Saliola, Mike Zabrocki
preprint arXiv:2208.07258
with Joseph Pappe, Digjoy Paul
European J. Comb. 108 (2023) 103640 ( arXiv:2204.06751 )
with Rosa Orellana, Franco Saliola, Mike Zabrocki
Algebraic Combinatorics 5 (2022) 5, 1165-1203 ( arXiv:2112:13909 )
with Stuart Margolis, John Rhodes
preprint arXiv:2110.10373
with Joseph Pappe, Digjoy Paul
Electronic J. Comb. 29(2) (2022), #P2.13
( arXiv:2109.06300 )
with Jianping Pan, Joseph Pappe, Wencin Poh
Annals of Combinatorics 26 (2022), 261-301 ( arXiv:2012.14975 )
with John Rhodes
Seminaire Lotharingien de Combinatoire 85B (2021) Article #27, 12 pp.
with John Rhodes
SIAM J. on Discrete Mathematics 36(4) (2022), 3031-3057 ( arXiv:2010.08879 )
with John Rhodes, Pedro V. Silva
International Journal of Algebra and Computation 32(3) (2022), 443-460 ( arXiv:2007.15552 )
with John Rhodes
Springer Proceedings in Mathematics & Statistics, vol 345. Springer, Singapore.
( arXiv:2006.00895 )
with Jennifer Morse, Jianping Pan, Wencin Poh
Electronic J. Comb. 27(2) (2020), #P2.29 ( arXiv:1911.08732 )
Crystal for Stable Grothendieck polynomials
Seminaire Lotharingien de Combinatoire 84B (2020) Article #33, 12 pp
with Eugene Gorsky, Graham Hawkes, Anne Schilling, Julianne Rainbolt
Algebraic Combinatorics 3(4) (2020) 855-886 ( arXiv:1905.10973 )
with Laura Colmenarejo, Rosa Orellana, Franco Saliola, Mike Zabrocki
J. Algebra 557 no. 1, (2020), 97-128 ( arXiv:1905.02071 )
An insertion algorithm for diagram algebras
Seminaire Lotharingien de Combinatoire 84B (2020) Article #48, 12 pp
with John Rhodes
International Journal of Algebra and Computation 29, no. 8, (2019) 1431-1449
( arXiv:1902.01042 )
with Wencin Poh
Seminaire Lotharingien de Combinatoire
82B (2019) #102, 11pp
with Maria Gillespie, Graham Hawkes, Wencin Poh
Bahar Acu et al. (eds.),
Advances in Mathematical Sciences,
Association for Women in Mathematics Series 21
with Maria Gillespie, Graham Hawkes, Wencin Poh
J. Combinatorial Theory, Series A 173 (2020) 105235 ( arXiv:1809.04647 )
with John Rhodes
Advances in Mathematics 347 (2019) 739-779 ( arXiv:1711.10689 )
with Georgia Benkart, Laura Colmenarejo, Pamela E. Harris, Rosa Orellana, Greta Panova, Martha Yip
Advances in Applied Math. 95 (2018) 96-115 ( arXiv:1707.08709 )
Seminaire Lotharingien de Combinatoire
80B (2018) #12, 12pp
with Masato Okado and Travis Scrimshaw
J. Algebra 516 (2018) 1-37 ( arXiv:1707.04876 )
with Sami Assaf
Algebraic Combinatorics 1(2) (2018) 225-247 ( arXiv:1705.09649 )
with Graham Hawkes and Kirill Paramonov
Seminaire Lotharingien de Combinatoire 78B (2017) #8, 12 pp
Electronic J. Comb. 24(3) (2017) #P3.51
( arXiv:1704.00889 )
with Daniel Bump
World Scientific, 2017,
ISBN: 978-981-4733-43-4, 292pp
MAA review of the book!
Errata
with Tobias Johnson and Erik Slivken
Electronic Journal of Probability 22 (2017), paper no. 18, 1-26 ( arXiv:1605.09777 [math.PR] )
with Cristian Lenart, Satoshi Naito, Daisuke Sagaki, Mark Shimozono
Rev. Roumaine Math. Pures Appl. 62 (2017), 1, 113-135
with John Rhodes and Pedro V. Silva
International Journal of Algebra and Computation 26(4) (2016) 675-704 ( arXiv:1604.00959 [math.GR] )
with Masato Okado, Reiho Sakamoto, Travis Scrimshaw
Journal of Algebraic Combinatoris 46 (2017) 341-401 ( arXiv:1603.08121 [math.QA] )
with Cristian Lenart, Satoshi Naito, Daisuke Sagaki, Mark Shimozono
Transformation Groups 22(4) (2017) 1041-1079 ( arXiv:1511.00465 [math.QA] )
with John Rhodes and Pedro V. Silva
International Journal of Algebra and Computation 26(4) (2016) 635-673 ( arXiv:1509.03383 [math.GR] )
with Nicolas M. ThiƩry, Graham White, Nathan Williams
European J. Combin. 62 (2017) 15-34 ( arXiv:1507.00656 [math.CO] )
with Cristian Lenart, Satoshi Naito, Daisuke Sagaki, Mark Shimozono
Toyama Math. Journal 37 (2015) 107-130
( arXiv:1504.04707 [math.QA] )
with Jennifer Morse
DMCTS proc FPSAC'15 (2015) 405-416
with Arvind Ayyer and Nicolas M. Thiery
Experimental Math. 26(1) (2017) 22-30
( arXiv:1412.7488 [math.PR] )
with Travis Scrimshaw
Electronic J.
Combinatorics 22(1) (2015) #P1.73
( arXiv:1409.2920 [math.CO] )
with Jennifer Morse
International Mathematics Research Notices 2016(8) (2016) 2239-2294
( arXiv:1408.0320 [math.CO] )
in: The Mathematical Legacy of Richard P. Stanley, AMS 2016, edited by P. Hersh, T. Lam, P. Pylyavskyy, V. Reiner, pp. 287-299 ( arXiv:1405.2966 [math.CO] )
with Cristian Lenart, Satoshi Naito, Daisuke Sagaki, and Mark Shimozono
International Mathematics Research Notices 2017(14) (2017) 4259-4319 ( arXiv:1402.2203 [math.QA] )
with Arvind Ayyer, Benjamin Steinberg, and Nicolas M. Thiery
International Journal of Algebra and Computation 25 (2015) 69-231
( arXiv:1401.4250 [math.CO] )
with Jennifer Morse
DMTCS proc. AT (2014) 489-500
with Cristian Lenart, Satoshi Naito, Daisuke Sagaki, and Mark Shimozono
Advanced Studies in Pure Mathematics (Mathematical Society of Japan, Tokyo) 71 (2016), Schubert Calculus - Osaka 2012, pp. 267-294
( arXiv:1308.3529 [math.QA] )
with Arvind Ayyer and Steven Klee
Fields Institute Communications
71 (2014) 285-304 (Springer)
( arXiv:1307.7499 [math.PR] )
with Arvind Ayyer, Benjamin Steinberg, Nicolas M. Thiery
Communications Math. Physics 335 (2015) 1065-1098
( arXiv.1305.1697 [math.PR] )
with Thomas Lam, Luc Lapointe, Jennifer Morse, Mark Shimozono, and Mike Zabrocki
Fields Institute Monographs 33 (Springer),
ISBN 978-1-4939-0682-6 (eBook)
( arXiv.1301.3569 [math.CO] )
with Cristian Lenart, Satoshi Naito, Daisuke Sagaki, and Mark Shimozono
DMTCS proc AS (2013) 25-36 ( arXiv.1211.6019 [math.QA] )
with Cristian Lenart, Satoshi Naito, Daisuke Sagaki, and Mark Shimozono
International Mathematics Research Notices
2015(7):1848-1901. ( arXiv.1211.2042 [math.QA] )
with Arvind Ayyer and Steven Klee
J. Algebraic Combinatorics,
39(4) (2014) 853-881
( arXiv:1205.7074 [math.CO] )
with Jennifer Morse
DMTCS proc
AR (2012) 735-744
( arXiv:1207.0786 [math.CO] )
with Masato Okado and Reiho Sakamoto
J. Algebraic Combinatorics,
37(3) (2013) 571-599
( arXiv:1109.3523 [math.QA] )
with Cristian Lenart
Mathematische Zeitschrift,
273(1) (2013) 401-426
( arXiv:1107.4169 [math.CO] )
DMTCS proc
AR (2012) 9-20 (FPSAC version)
with Patricia Hersh
International Mathematics Research Notices 2013 (2) 463-473
( arXiv:1107.4073 [math.CO] )
with Peter Tingley
The Electronic Journal of Combinatorics 19(2) (2012) P2
( arXiv:1104.2359 [math.QA] )
DMTCS proc
AO (2011) 861-872 (FPSAC version)
with Florent Hivert and Nicolas Thiery
Algebra and Number Theory Journal 7(3) (2013) 595-671
( arXiv:1012.1361 [math.CO] )
with Tom Denton, Florent Hivert, Nicolas Thiery
Seminaire Lotharingien de Combinatoire, B64d (2011), 44 pp
( arXiv:1010.3455 [math.RT] )
with Jason Bandlow and Mike Zabrocki
Journal of Combinatorial Theory, Series A, 118(5) (2011) 1588-1607
( arXiv:1004.4886 [math.C0] )
DMTCS proc
AO (2011) 99-110 (FPSAC version)
Oberwolfach Report
with Florent Hivert and Nicolas Thiery
DMTCS proc
AN (2010) 307-318 ( arXiv:0912.2212 [math.CO] )
with Brant Jones
J. Algebra 324 (2010) 2512-2542
( arXiv:0909.2442 [math.CO] )
DMTCS proc
AN (2010) 785-796 (FPSAC version)
with Qiang Wang
The Electronic Journal of Combinatorics 17 Issue 1 (2010) R24
( arXiv:0908.2458 [math.CO] )
with Thomas Lam and Mark Shimozono
Compositio Mathematica 146 Issue 4 (2010) 811-852 ( arXiv:0901.1506 [math.CO] )
with Ghislain Fourier and Masato Okado
Contemp. Math. 506 (2010) 127-143 ( arXiv:0811.1604 [math.RT] )
DMTCS proc AK (2009) 426-439 (FPSAC version)
with Ghislain Fourier and Masato Okado
Advances in Mathematics
222 Issue 3 (2009) 1080-1116
( arXiv:0810.5067 [math.RT] )
with Jason Bandlow and Nicolas Thiery
J. Algebraic Combinatorics
31 (2010) 217-251
( arXiv:0806.3131 [math.CO] )
Journal of Combinatorial Theory, Series A 116 (2009) 844-863
( arXiv:0804.3781 [math.RT] )
Hecke group algebras as degenerate affine Hecke algebras
DMTCS proc AJ (2008) 611-624
with Florent Hivert and Nicolas Thiery
with Thomas Lam and Mark Shimozono
Mathematische Zeitschrift 264(4) (2010) 765-811
( arXiv:0710.2720 [math.CO] )
with Jean-Christophe Novelli
RIMS Kokyuroku Bessatsu B8 (2008) 71-83
( arXiv:0706.2996 [math.CO] )
with Masato Okado
Representation Theory 12 (2008) 186-207
( arXiv:0706.2224 [math.QA] )
J. Algebra
319 (2008) 2938-2962
( arXiv:0704.2046 [math.QA] )
with Ghislain Fourier and Mark Shimozono
J. Algebra
309 (2007) 386-404
( math.QA/0605451 )
MSJ Memoirs 17 (2007) 75-104 (published by the Mathematical Society of Japan)
( math.QA/0512161 )
Formal Power Series and Algebraic Combinatorics, San Diego, California 2006
( pdf file )
with Lipika Deka
J. Combinatorial Theory, Series A
113 (2006) 1435-1461
( math.CO/0509194 )
International Mathematics Research Notices, Volume 2006,
Article ID 97376, Pages 1-27
( math.QA/0508107 )
with Lipika Deka
Commun. Math. Phys. 260 (2005) 711-725
( math-ph/0412084 )
with Mark Shimozono
J. Algebra 295 (2006) 562-610
( math.QA/0412376 )
with Philip Sternberg
in: A. Kuniba, M. Okado (Eds.), RIMS Kokyuroku 1429
"Combinatorial Aspect of Integrable Systems", to appear.
( postscript file )
with Philip Sternberg
J. Alg. Combin. 23 (2006) 317-354
( math.QA/0408113 )
J. Algebra 285 (2005) 292-334
( math.QA/0406248 )
Supplementary Notes
in: B. Lulek, T. Lulek and A. Wal (Eds), "Symmetry and Structural Properties
of Condensed Matter", Vol. 7, World Scientific, Singapore 2003; pp. 201-224
( math-ph/0210014 )
with Masato Okado and Mark Shimozono
Commun. Math. Phys.
238 (2003) 187-209
( math.QA/0209082 )
with Masato Okado and Mark Shimozono
"Algebraic Combinatorics and Quantum Groups", Edited by N. Jing, World Scientific (2003), 85-124
( math.QA/0203163 )
with Masato Okado and Mark Shimozono
J. Algebra 267 (2003) 212-245
( math.QA/0111288 )
in: MathPhys Odyssey 2001, M. Kashiwara and T. Miwa (eds.),
Birkhaeuser Boston, Cambridge, MA, 2002, pp. 437-454
( math.CO/0107214 )
with Masato Okado and Mark Shimozono
Representation Theory
7 (2003) 101-163
( math.QA/0105017 )
with Masato Okado and Mark Shimozono
Contemp. Math. 291 (2001) 29-53
( math.QA/0104268 )
with Mark Shimozono and Dennis E. White
Advances in Applied Mathematics 30 (2003) 258-272
( postscript file )
with Mark Shimozono
Commun. Math. Phys. 220 (2001) 105-164
( math.QA/0001114 )
with Mark Shimozono
in: Formal Power Series and Algebraic Combinatorics,
D. Krob, A.A. Mikhalev, A.V. Mikhalev (eds.),
12th International conference, FPSAC'00, Moscow, Russia, June 2000,
Proceedings, Springer Verlag Berlin/Heidelberg 2000, pp. 367-378.
( postscript file )
with S. Ole Warnaar
in: Physical Combinatorics, M. Kashiwara and T. Miwa (eds.),
Birkhaeuser Boston, Cambridge, MA, 2000, pp. 163-183.
( math.QA/9909044 )
with S. Ole Warnaar
Contemp. Math. 297 (2002) 227-255
( math.QA/9906092 )
with Anatol N. Kirillov and Mark Shimozono
Selecta Mathematica (N.S.) 8 (2002) 67-135
( math.CO/9901037 )
with Anatol N. Kirillov and Mark Shimozono
Seminaire Lotharingien de
Combinatoire 42 [B42j] (1999)
( postscript file )
with Mark Shimozono
Advanced Studies in Pure Mathematics 28,
Combinatorial Methods in Representation Theory (2000) 305-325
( math.QA/9812106 )
with George E. Andrews and S. Ole Warnaar
J. Amer. Math. Soc. 12 (1999) 677-702
( math.QA/9807125 )
with S. Ole Warnaar
Commun. Math. Phys. 202 (1999) 359-401
( math.QA/9802111 )
with Alexander Berkovich, Barry M. McCoy, and S. Ole Warnaar
Nucl. Phys. B499 [PM] (1997) 621-649
( hep-th/9702026 )
with S. Ole Warnaar
The Ramanujan Journal 2 (1998) 459-494
( q-alg/9701007 )
with Alexander Berkovich and Barry M. McCoy
Commun. Math. Phys. 191 (1998) 325-395
( q-alg/9607020 )
with S. Ole Warnaar
The Ramanujan Journal 2 (1998) 327-349
( q-alg/9607014 )
with S. Ole Warnaar
Int. J. of Mod. Phys. B11 (1997) 189-195
( q-alg/9604015 )
with Alexander Berkovich and Barry M. McCoy
Physica A 228 (1996) 33-62
( hep-th/9512182 )
Nucl. Phys. B467 (1996) 247-271
( hep-th/9510168 )
Nucl. Phys. B459 (1996) 393-436
( hep-th/9508050 )
with Peter van Nieuwenhuizen
Phys. Rev. D50 (1994) 967-970
( hep-th/9408125 )