My research interests are mostly in algebraic combinatorics. Lately I have been studying (a) the representation theory of the symmetric group by understanding the characters as symmetric functions (b) Symmetric function problems like plethysm and the restriction problem (how a \$Gl_n\$ irreducible decomposes as an \$S_n\$ irreducible) (c) representations of partition algebras and a new multiset partition algebra (d) \$q,t\$-combinatorics, symmetric functions and the corresponding representation theory My Google scholar page.

## Published papers

A Macdonald Vertex Operator and Tableaux Statistics for the Two-Column (q,t)-Kostka Coefficients, Electronic Journal of Combinatorics, 5, R45, (extended abstract)
Vertex operators for standard bases of the symmetric functions, J. of Alg. Comb., 13, No. 1 (2000), pp. 83-101. (arXiv:math.CO/9904084)
Positivity for special cases of (q,t)-Kostka coefficients and standard tableaux statistics, Electronic Journal of Combinatorics, 6, R41
(with Mark Shimozono) Hall-Littlewood vertex operators and generalized Kostka polynomials, Adv. in Math., 158 (2001), pp. 66-85.  (arXiv:math.QA/0001168)
Ribbon Operators and Hall-Littlewood Symmetric Functions, Adv. in Math., 156 (2000), pp. 33-43. (arXiv:math.QA/0008163) (extended abstract)
(with A. Garsia) Polynomiality of the q,t-Kostka Revisited, Algebraic combinatorics and computer science, Springer Italia, Milan, (2001), pp. 473-491. (arXiv:math.QA/0008199)
q-Analogs of symmetric function operators, Disc. Math., 256 (2002), pp. 831-853. (arXiv:math.QA/0008188)
(with N. Bergeron) q and q,t-Analogs of Non-commutative Symmetric Functions, Discrete Mathematics, 298, 1, (2005), pp. 79-103. (math.CO/0106255)
(with G. Tudose) q-Analogs of Q-Schur functions, Algebraic Combinatorics and Quantum Groups, Nihuan Jing, ed., World Sci. Publ., 2003. (math.CO/0203046) (extended abstract)
(with C. De Concini, C. Procesi, E. Rogora) On Equations of Degree 3 * 2^m, Boletin del Instituto de Matematica ``Beppo Levi,'' Vol II, Rosario, Argentina, (2003/4), pp. 109-122. (pdf version)
(with M. Shimozono) Deformed universal characters for classical and affine algebras, Journal of Algebra, 299 (2006), pp. 33-61, (arXiv:math.CO/0404288) (extended abstract)
(with M. H. Albert, M. Elder, A. Rechnitzer, P. Westcott) On the Wilf-Stanley limit of 4231-avoiding permutations and a conjecture of Arratia, Adv. in Applied Mathematics, 36, 2, (2006), pp. 96-105. (arXiv:math.CO/0502504)
(with N. Bergeron, C. Hohwleg) Posets related to the connectivity set of Coxeter groups, J. of Alg., 303, 2, (2006), pp. 831-846, (math.CO/0509271).
(with N. Bergeron, C. Hohwleg, and M. Rosas) Grothendieck bialgebras, Partition lattices and symmetric functions in noncommutative variables, Electr. J. of Comb., R75, Volume 13(1), 2006, 19 pages.
(with N. Bergeron, C. Reutenauer, M. Rosas) Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, Canadian Math Journal (2008), Vol 60, No 2, pp. 266-296. (arXiv:math.CO/0502082)
(with E. Briand, M. Rosas) On the S_n-module structure of the noncommutative harmonics, Journal of Combinatorial Theory, Series A, Volume 115, Issue 6, August 2008, pages 1077-1085. (arXiv:0704.1101)
(with A. Garsia, N. Wallach, G. Xin) Hilbert Series of Invariants, Constant terms and Kostka-Foulkes Polynomials, Discrete Mathematics, Volume 309, Issue 16, pp. 5206-5230 (2009). (pdf version)
(with N. Bergeron, F. Descouens) A non-commutative generalization of k-Schur functions, Discrete Mathematics, Volume 309, Issue 16, pp. 5092-5105 (2009). (arXiv:0804.0944v1)
(with N. Bergeron, F. Descouens) A Filtration of (q,t)-Catalan numbers, Advances in Applied Mathematics 44 pp. 1636 (2010) . (arXiv:0806.3046)
(with N. Bergeron) The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and cofree, J. of Alg. and Its Appl., Volume: 8 (4), pp. 581-600 (2009) . (arXiv:math.CO/0509265)
(with A. Brown, S. van Willigenburg) Expressions for Catalan Kronecker Products, Pacific Journal of Mathematics, Vol. 248 (2010), No. 1, 31-48. (arXiv:0809.3469)
(with J. Bandlow and A. Schilling) The Murnaghan-Nakayama Rule for k-Schur Functions, Journal of Combinatorial Theory, Series A Volume 118, Issue 5, June 2011, pp. 1588-1607, (arXiv:1004.4886)
(with A. Garsia and G. Xin) Hall-Littlewood Operators in the Theory of Parking Functions and Diagonal Harmonics, Int Math Res Notices 2012 (6): 1264-1299. doi: 10.1093/imrn/rnr060
(with J. Haglund and J. Morse) A compositional shuffle conjecture specifying touch points of the Dyck path, (arXiv:1008.0828v2), Canadian Journal of Mathematics 64(2012), no. 4, 822-844. doi:10.4153/CJM-2011-078-4.
(with Marcelo Aguiar, Carlos Andre, Carolina Benedetti, Nantel Bergeron, Zhi Chen, Persi Diaconis, Anders Hendrickson, Samuel Hsiao, I. Martin Isaacs, Andrea Jedwab, Kenneth Johnson, Gizem Karaali, Aaron Lauve, Tung Le, Stephen Lewis, Huilan Li, Kay Magaard, Eric Marberg, Jean-Christophe Novelli, Amy Pang, Franco Saliola, Lenny Tevlin, Jean-Yves Thibon, Nathaniel Thiem, Vidya Venkateswaran, C. Ryan Vinroot, Ning Yan) Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras (arXiv:1009.4134), Advances in Mathematics, Volume 229, Issue 4, 1 March 2012, Pages 2310--2337.
(with A. Bergeron-Brlek and C. Hohlweg) Words and polynomial invariants of finite groups in non-commutative variables, (arXiv:0907.0814), Annals of Combinatorics: Volume 16, Issue 1 (2012), Page 1-36, DOI: 10.1007/s00026-011-0119-0.
(with A. Garsia, N. Wallach, G. Xin) Kronecker Coefficients via Symmetric Functions and Constant Term identities, (pdf version), International Journal of Algebra and Computation, Volume No. 22, Issue No. 3 (2012). doi: 10.1142/S0218196712500221
(with C. Berg, N. Bergeron and H. Thomas) Expansion of k-Schur functions for maximal k-rectangles within the affine nilCoxeter algebra, Journal of Combinatorics, Volume 3, Number 3, 563--589, 2012, (arXiv:1107.3610).
(with A. Duane and A. Garsia) A new dinv Arising from the Two Part Case of the Shuffle Conjecture, Journal of Algebraic Combinatorics, Volume 37, Issue 4 (2013), Page 683-715 (DOI: 10.1007/s10801-012-0382-0) (arXiv:1205.6128).
(with C. Berg, N. Bergeron, S. Pon) Expansions of k-Schur functions in the affine nilCoxeter algebra, (arXiv:1111.3588v2) v19i2p55 Electronic Journal of Combinatorics.
(with A. Garsia, G. Xin) Proof of the 2-part Compositional Shuffle Conjecture, pdf, published in Progress in Mathematics (Book 257), Birkhauser; 2014, 538 pages, pp. 227-257.
(with C. Berg, N. Bergeron, F. Saliola, L. Serrano) A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions, Canadian Journal of Mathematics 66(2014), no. 3, 525-565. http://dx.doi.org/10.4153/CJM-2013-013-0, (arXiv:1208.5191), Canad. J. Math. 66(2014), 525--565.
(with C. Berg, N. Bergeron, F. Saliola, L. Serrano) The immaculate basis of the non-commutative symmetric functions, (arXiv:1303.4801), FPSAC 2013 extended abstract (this is a summary of some results in arXiv:1208.5191, arXiv:1304.1224 and arXiv:1305.4700).
(with C. Berg, N. Bergeron, F. Saliola, L. Serrano) Indecomposable modules for the dual immaculate basis of quasi-symmetric functions, Proceedings of the AMS, Volume 143, Number 3, March 2015, Pages 991-1000. (arXiv:1304.1224).
(with A. Garsia and G. Xin) A three shuffle case of the compositional parking function conjecture, Journal of Combinatorial Theory, Series A 123/1 (2014), pp. 202-238 (doi: 10.1016/j.jcta.2013.12.008), (arXiv:1208.5796).
(with A. Garsia, J. Haglund, G. Xin) Some new applications of the Stanley-Macdonald Pieri Rules, for , in proceedings for for Stanley@70 (MIT, June 23-27, 2014), The Mathematical Legacy of Richard P. Stanley, AMS, (2017) pp. 1--25. (arXiv:1407.7916).
(with N. Bergeron, Juana Sánchez-Ortega) The Pieri rule for dual immaculate quasi-symmetric functions, (arXiv:1307.4273), Annals of Combinatorics, June 2016, Volume 20, Issue 2, pp 283--300. doi:10.1007/s00026-016-0303-3. (arXiv:arXiv:1307.4273)
(with C. Berg and N. Williams) Symmetries of the k-bounded partition lattice, (arXiv:1111.2783v3), Annals of Combinatorics, June 2016, Volume 20, Issue 2, pp 251--281. doi:10.1007/s00026-016-0307-z.
(with C. Berg, N. Bergeron, F. Saliola, L. Serrano) Multiplicative structures of the immaculate basis of non-commutative symmetric functions, (arXiv:1305.4700), Journal of Combinatorial Theory, Series A Volume 152, November 2017, Pages 10--44, https://doi.org/10.1016/j.jcta.2017.05.003.
A proof of the 4-variable Catalan polynomial of the Delta conjecture, Journal of Combinatorics, Volume 10 (2019), Number 4, pp. 599--632, DOI: http://dx.doi.org/10.4310/JOC.2019.v10.n4.a1. issue in honor of Jeff Remmel, (arXiv:1609.03497).
(with R. Orellana) Products of symmetric group characters, Journal of Combinatorial Theory, Series A, Volume 165, July 2019, Pages 299--324. https://doi.org/10.1016/j.jcta.2019.02.019 (arXiv:1709.08098). Worksheets that compute the combinatorial coefficients described in the paper: pdf and jupyter notebook format.
(with L. Colmenarejo, R. Orellana, F. Saliola, A. Schilling) An insertion algorithm on multiset partitions with applications to diagram algebras, Journal of Algebra, Volume 557, 1 September 2020, Pages 97--128. https://doi.org/10.1016/j.jalgebra.2020.04.010 (arXiv:1905.02071).
(with R. Orellana) A combinatorial model for the decomposition of multivariate polynomials rings as an \$S_n\$-module, Electronic Journal of Combinatorics, Issue Volume 27, Issue 3 (2020), P3.24, (https://doi.org/10.37236/8935) (arXiv:1906.01125).
(with R. Orellana) The Hopf structure of symmetric group characters as symmetric functions, Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 551--574. (https://doi.org/10.5802/alco.170) (arXiv:1901.00378).
(with R. Orellana) Symmetric group characters as symmetric functions, (arXiv:1605.06672), also extended abstract, (arXiv:1510.00438), Advances in Mathematics, Volume 390, 29 October 2021, 107943. (https://doi.org/10.1016/j.aim.2021.107943).
(with R. Orellana, F. Saliola, A. Schilling) Plethysm and the algebra of uniform block permutations, Alg. Comb., Volume 5 (2022), no. 5, pp. 1165--1203, (doi:10.5802/alco.243) (arXiv:2112.13909).
(with R. Orellana) Howe duality of the symmetric group and a multiset partition algebra, Communications in Algebra, Volume 51 (2023), no. 1, pp. 393--413. (doi:10.1080) (arXiv:2007.07370).
(with N. Bergeron, Kelvin Chan, Farhad Soltani) Quasisymmetric harmonics of the exterior algebra, Canadian Mathematical Bulletin, 2023, pp. 1--17. (doi:10.4153) (arXiv:2206.02065) .
(with R. Orellana) The Symmetric Group Through a Dual Perspective, Not. of the AMS, Volume 70, Number 6, pp. 897--904. (DOI: 10.1090/noti2704) (also cover art)
(with R. Orellana and Nancy Wallace) Representations of the quasi-partition algebras (arXiv:2306.17326), Journal of Algebra, 2024 (DOI: j.jalgebra.2024.01.025).
(with L. Colmenarejo, R. Orellana, F. Saliola, A. Schilling) The mystery of plethysm coefficients, (arXiv:2208.07258), to appear in Proceedings of Symposia in Pure Mathematics for the Open Problems in Algebraic Combinatorics 2022 conference.

## Other Papers/Projects/Drafts/Submissions

the 'drafts' are all in a nearly finished form, if you want a copy please contact me and I can send you the current version
(with R. Orellana) Some remarks on the characters of the general Lie superalgebra, (arXiv:math.CO/0008152)
A Bijective Proof of an Unusual Symmetric Group Generating Function (arXiv:math.CO/0310301)
Some Calculations on the Outer Hookwalk - notes for my advancement to candidacy in 1994(?) (notes)
(with M. Mishna, draft) Analytic aspects of the shuffle product, Dans Proceedings of the 25th Annual Symposium on the Theoretical Aspects of Computer Science - STACS 2008, Bordeaux : France (2008) (arXiv:0802.2844)
A module for the Delta conjecture, (arXiv:1902.08966).
(with R. Orellana, F. Saliola, A. Schilling) From quasi-symmetric to Schur expansions with applications to symmetric chain decompositions and plethysm (arXiv:2404.04512)
(with R. Orellana, F. Saliola, A. Schilling) The lattice of submonoids of the uniform block permutations containing the symmetric group (arXiv:2405.09710)

## Books

(with T. Lam, L. Lapointe, J. Morse, A. Schilling, M. Shimozono) Book: k-Schur functions and affine Schubert calculus, Fields Institute Monographs, Vol 33, Springer (2014), 219 p. (arXiv:1301.3569).
Mini Mathematical Monographs 1: An Introduction to Ordinary Generating Functions (Self Published).

## Talks and posters

PDF of slides for talk "q-Analogs of Hopf Algebra Structures" given in Kyoto Aug '01
PDF of poster presented at FPSAC'01 in Scottsdale, Arizona June '01 "The ribbon rule for homogeneous symmetric functions"  (extended abstract)
PDF of slides for talk "A quantization of non-commutative symmetric functions"  at the CMS Winter meeting '01 and updated for a seminar at UWaterloo June, '02. This is similar to the chalk talk that I presented at several combinatorics seminars (York, UMinnesota, UQaM) Fall '01.
PDF of slides "Ribbons and Column Strict Tableaux" presented at the Univeristy of Miami and York University colloquium Feb '02.
PDF of slides "A q-analog of Schur's Q-functions" presented at the Wilfrid Laurier University April '02. Description of this research.
FPSAC '02. "Hall-Littlewood analogs in the Q-function algebra" (PDF slides) and "q and qt analogs of non-commutative symmetric functions." (PDF slides)
Prof Talk- for Club Infinity at York University. "Gems of algebra: The secret life of the symmetric group" (PDF of slides)
A Java program to play a game of extended "Set."  I learned a lot about the work that has been done on the game of "Set" at the FPSAC '01 conference where I gave a demo of this program, but I have not had the chance to rewrite the material on the web page to reflect that.
A gallery of posets.  Pictures of permutahedron and other geometric shapes arising from posets.
A series of three talks on symmetric functions in non-commuting variables that I gave in Rome, December 2006 for the Incontro Italiano di Combinatoria Algebrica.
A zoo of Hopf algebras
Combinatorics and coinvariants
A representation theoretical approach
Slides from two talks that I gave at the meeting Combinatorial Algebra meets Algebraic Combinatorics.
A (nother version of the) zoo of Hopf algebras
The wilderness of Hopf algebras
Presentation that I gave at Wake Forest University (updated from the one that I gave at the Fields Institute)
k-Schur functions indexed by a maximal rectangle
A generalization of Suter symmetry for k-bounded partitions, poster where the physical copy was lost somewhere in Japan
Find me a basis that... for Sage Days 49. The goal I had for this talk was to present some ideas that Sage projects and what I wanted to use Sage for. I introduced here the concept of a '(super) totally awesome' basis for a Hopf algebra. Question: what are the (super) totally awesome bases of (N/Q)Sym? Is this within our computational power?
Introduction to Sage or How I learned to stop worrying and love py-thon - An introduction to Sage talk - what Sage does well (and not so well) and why you should (or shouldn't) migrate from other computer algebra systems
Slides from talks on the irreducible character basis:
CMS Winter meeting 2015, On symmetric group and partition algebra characters,
Sage Days 78 2016, Open Problems in Combinatorial Representation Theory,
Coloquio Latinoamericano de Algebra 2016, Irreducible characters of the symmetric group as symmetric functions,
AMS Sectional meeting in Minneapolis on Combinatorial representation theory Oct 2016, Pieri rules and combinatorics of symmetric group characters,
Algčbre et combinatoire au LaCIM, September 2018, The representation theory of the symmetric group revisited.
YouTube videos of talks on symmetric group characters:
IMSc Algebraic Combinatorics Seminar, Chennai, online (India), May 2020
IPAC 2023:
Talk 1 (Rosa) - Schur-Weyl duality, (reduced) Kronecker, Partition Algebra - video - slides
Talk 2 (Mike) - representation theory of S_n, restriction problem - video - slides - notes
Talk 3 (Mike) - better definitions of the character basis - video - slides - notes
Talk 4 (Rosa) - change of bases - multiset partitions and tableaux - video - slides
Talk 5 (Mike) - combinatorial interpretations for some reduced Kronecker - video - slides
Talk 6 (Rosa) - more change of bases, products and combinatorial interpretations - video - slides
An extension of RSK to multi-set partitions:
Dartmouth colloquium, From Robinson-Schensted-Knuth correspondence to Schur-Weyl duality,
Coloquio Latinoamericano de Algebra 2017, Combinatorics of characters of symmetric group as symmetric functions
CMS Winter meeting 2017, Symmetric group representations and Howe duality
AMS Nashville Apr 2018, A multiset partition algebra,
Oberwolfach Feb 2020, The multiset partition algebra and Kronecker product

Last Updated: May 15, 2024