Professor Andrew Mathas

F07 - Carslaw Building
The University of Sydney

Telephone 9351 6058
Fax 9351 4534

Website Personal web page

Research interests

The representation theory of the symmetric groups, and the closely related cyclotomic Hecke algebras of type A, was transformed when Brundan and Kleshchev's discovered of a grading on these algebras following work of Khovanov-Lauda and Rouquier. The graded theory is harder than the "classical" approach to this subject but it reveals new features of the representation theory which we could not see before.

The underlying problems are still the same - we want to compute the (graded) dimensions and the (graded) decomposition numbers of these algebras - but there is now more structure to work with. There are indications that this new perspective may furnish us with the tools to finally answer these questions.

This theory is intimately connected with the representation theory of affine Hecke algebras and quantum groups; there are also ramifications for the representation theory of the symmetric groups and finite reductive groups.

Other active interests include:

  • Quiver Hecke algebras and quiver Schur algebras
  • Cyclotomic Hecke algebras, complex reflection groups and their braid groups.
  • The q-Schur algebras and cyclotomic q-Schur algebras.
  • Combinatorics of symmetric groups and Hecke algebras.
  • The theory of (graded) cellular algebras.
  • Affine Hecke algebras.
  • Quantum groups, canonical bases, and crystal graphs.
  • Kazhdan-Lusztig polynomials and cell representations.
  • Coxeter groups and groups of Lie type, and their representation theory.

Teaching and supervision

Timetable

Awards and honours

  • 2009-13 Australian Professorial Fellow, Australian Research Council.
  • 2007 Fellow of the Australian Mathematical Society.
  • 2006 Medal of the Australian Mathematical Society.
    (Awarded annually for distinguished research in the mathematical
    sciences to an Australian mathematician under 40.)
  • 2004 Faculty of Science Teaching Award, University of Sydney.
  • 2004 JSPS Invitation Fellowship, Research institute for Mathematical Sciences, Kyoto, Japan.
  • 2003 EPSRC Visiting Research Fellow, Imperial College, London.
  • 2001 EPSRC Visiting Research Fellow, University of Leicester.
  • 1997-00 U2000 Postdoctoral Fellowship, University of Sydney.

Selected grants

2015

  • The dimension problem for Hecke algebras; Mathas A; Australian Research Council (ARC)/Discovery Projects (DP).

2011

  • Graded representations of Hecke algebras; Mathas A, Brundan J; Australian Research Council (ARC)/Discovery Projects (DP).

2010

  • Graded decomposition matrices for symmetric groups; Mathas A, Kleshchev A; DVC Research/International Visiting Research Fellowship (IVRF).

2009

  • Pyramids and decomposition numbers for the symmetric and general linear groups; Mathas A, Henderson A; Australian Research Council (ARC)/Discovery Projects (DP).

2008

  • Modular representations of cyclotomic algebras; Mathas A; Australian Research Council (ARC)/Discovery Projects (DP).

2007

  • Modular representations of cyclotomic algebras; Mathas A; University of Sydney/Bridging Support.

2006

  • Affine buildings and Hecke algebras; Mathas A; Australian Research Council (ARC)/Discovery Projects (DP).

2004

  • Japanese Invitation fellowship, Japanese Society for the promotion of Science; Mathas A; Japanese Society for the Promotion of Science/Fellowship.
  • Induction functors for cyclotomic algebras; Mathas A; DVC Research/Research and Development Scheme: Research and Development (R&D).

2003

  • Quantized representation theory; Mathas A; Australian Research Council (ARC)/Discovery Projects (DP).

2001

  • Decomposition numbers of symmetric groups; Mathas A; DVC Research/Research and Development Scheme: Newly Appointed Staff (NAS).

2000

  • Cyclotomic representation theory.; Mathas A; Australian Research Council (ARC)/Small Grants.

Selected publications

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Journals

  • Lyle, S., Mathas, A. (2014). Cyclotomic Carter-Payne homomorphisms. Representation Theory, 18(June 2014), 117-154.
  • Mathas, A., Soriano, M. (2013). Blocks of the truncated q-Schur algebras of type A. Contemporary Mathematics, 602, 123-141. [More Information]
  • Hu, J., Mathas, A. (2012). Decomposition numbers for Hecke algebras of type G(r, p, n): the (ε, q)-separated case. Proceedings of the London Mathematical Society, 104(5), 865-926. [More Information]
  • Hu, J., Mathas, A. (2012). Graded induction for Specht modules. International Mathematics Research Notices, 2012 (6), 1230-1263. [More Information]
  • Kleshchev, A., Mathas, A., Ram, A. (2012). Universal graded Specht modules for cyclotomic Hecke algebras. Proceedings of the London Mathematical Society, 105(3), 1245-1289. [More Information]
  • Lyle, S., Mathas, A. (2010). Carter-Payne homomorphisms and Jantzen filtrations. Journal of Algebraic Combinatorics, 32(3), 417-457. [More Information]
  • Hu, J., Mathas, A. (2010). Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A. Advances in Mathematics, 225(2), 598-642. [More Information]
  • Mathas, A. (2009). A Specht filtration of an induced Specht module. Journal of Algebra, 322(3), 893-902. [More Information]
  • Hu, J., Mathas, A. (2009). Morita equivalences of cyclotomic Hecke algebras of type G(r, p, n). Journal fuer die Reine und Angewandte Mathematik: Crelle's journal, 628, 169-194. [More Information]
  • Mathas, A., Orellana, R. (2008). Cyclotomic Solomon algebras. Advances in Mathematics, 219(2), 450-487. [More Information]
  • Mathas, A., Soriano, M. (2008). Seminormal forms and Gram determinants for cellular algebras. Journal fuer die Reine und Angewandte Mathematik: Crelle's journal, 619, 141-173.
  • Lyle, S., Mathas, A. (2007). Blocks of cyclotomic Hecke algebras. Advances in Mathematics, 216(2), 854-878. [More Information]
  • Ariki, S., Mathas, A., Rui, H. (2006). Cyclotomic Nazarov-Wenzl Algebras. Nagoya Mathematical Journal, 182, 47-134.
  • James, G., Lyle, S., Mathas, A. (2006). Rouquier blocks. Mathematische Zeitschrift, 252(3), 511-531. [More Information]
  • Ariki, S., Mathas, A. (2005). Corrigendum to "The representation type of Hecke algebras of type B" [Adv. Math. 181 (2004) 134-159]. Advances in Mathematics, 192, 228-230. [More Information]
  • Kunzer, M., Mathas, A. (2005). Elementary divisors of Specht modules. European Journal of Combinatorics, 26(6), 943-964. [More Information]
  • Lyle, S., Mathas, A. (2005). Row and column removal theorems for homomorphisms of Specht modules and Weyl modules. Journal of Algebraic Combinatorics, 22(2), 151-179. [More Information]
  • Mathas, A. (2004). Matrix units and generic degrees for the Ariki-Koike algebras. Journal of Algebra, 281(2), 695-730. [More Information]
  • James, G., Mathas, A. (2004). Symmetric Group Blocks Of Small Defect. Journal of Algebra, 279(2), 566-612. [More Information]
  • Ariki, S., Mathas, A. (2004). The representation type of Hecke algebras of type B. Advances in Mathematics, 181(1), 134-159.
  • Mathas, A. (2003). Tilting modules for cyclotomic Schur algebras. Journal fuer die Reine und Angewandte Mathematik: Crelle's journal, 562, 137-169.
  • James, G., Mathas, A. (2002). Equating decomposition numbers for different primes. Journal of Algebra, 258(2), 599-614.
  • Dipper, R., Mathas, A. (2002). Morita equivalences of Ariki-Koike algebras. Mathematische Zeitschrift, 240(3), 579-610. [More Information]

Conferences

  • Mathas, A., Ariki, S. (2004). Hecke Algebras With A Finite Number Of Indecomposable Modules. Mathematical Society of Japan's 10th International Conference : Representation Theory of Algebraic Groups and Quantum Groups, Tokyo, Japan: Mathematical Society of Japan.
  • Mathas, A. (2004). The Representation Theory Of The Ariki-Koike Algebras And The Cyclotomic q-Schur Algebras. Mathematical Society of Japan's 10th International Conference : Representation Theory of Algebraic Groups and Quantum Groups, Tokyo, Japan: Mathematical Society of Japan.

2014

  • Lyle, S., Mathas, A. (2014). Cyclotomic Carter-Payne homomorphisms. Representation Theory, 18(June 2014), 117-154.

2013

  • Mathas, A., Soriano, M. (2013). Blocks of the truncated q-Schur algebras of type A. Contemporary Mathematics, 602, 123-141. [More Information]

2012

  • Hu, J., Mathas, A. (2012). Decomposition numbers for Hecke algebras of type G(r, p, n): the (ε, q)-separated case. Proceedings of the London Mathematical Society, 104(5), 865-926. [More Information]
  • Hu, J., Mathas, A. (2012). Graded induction for Specht modules. International Mathematics Research Notices, 2012 (6), 1230-1263. [More Information]
  • Kleshchev, A., Mathas, A., Ram, A. (2012). Universal graded Specht modules for cyclotomic Hecke algebras. Proceedings of the London Mathematical Society, 105(3), 1245-1289. [More Information]

2010

  • Lyle, S., Mathas, A. (2010). Carter-Payne homomorphisms and Jantzen filtrations. Journal of Algebraic Combinatorics, 32(3), 417-457. [More Information]
  • Hu, J., Mathas, A. (2010). Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A. Advances in Mathematics, 225(2), 598-642. [More Information]

2009

  • Mathas, A. (2009). A Specht filtration of an induced Specht module. Journal of Algebra, 322(3), 893-902. [More Information]
  • Hu, J., Mathas, A. (2009). Morita equivalences of cyclotomic Hecke algebras of type G(r, p, n). Journal fuer die Reine und Angewandte Mathematik: Crelle's journal, 628, 169-194. [More Information]

2008

  • Mathas, A., Orellana, R. (2008). Cyclotomic Solomon algebras. Advances in Mathematics, 219(2), 450-487. [More Information]
  • Mathas, A., Soriano, M. (2008). Seminormal forms and Gram determinants for cellular algebras. Journal fuer die Reine und Angewandte Mathematik: Crelle's journal, 619, 141-173.

2007

  • Lyle, S., Mathas, A. (2007). Blocks of cyclotomic Hecke algebras. Advances in Mathematics, 216(2), 854-878. [More Information]

2006

  • Ariki, S., Mathas, A., Rui, H. (2006). Cyclotomic Nazarov-Wenzl Algebras. Nagoya Mathematical Journal, 182, 47-134.
  • James, G., Lyle, S., Mathas, A. (2006). Rouquier blocks. Mathematische Zeitschrift, 252(3), 511-531. [More Information]

2005

  • Ariki, S., Mathas, A. (2005). Corrigendum to "The representation type of Hecke algebras of type B" [Adv. Math. 181 (2004) 134-159]. Advances in Mathematics, 192, 228-230. [More Information]
  • Kunzer, M., Mathas, A. (2005). Elementary divisors of Specht modules. European Journal of Combinatorics, 26(6), 943-964. [More Information]
  • Lyle, S., Mathas, A. (2005). Row and column removal theorems for homomorphisms of Specht modules and Weyl modules. Journal of Algebraic Combinatorics, 22(2), 151-179. [More Information]

2004

  • Mathas, A., Ariki, S. (2004). Hecke Algebras With A Finite Number Of Indecomposable Modules. Mathematical Society of Japan's 10th International Conference : Representation Theory of Algebraic Groups and Quantum Groups, Tokyo, Japan: Mathematical Society of Japan.
  • Mathas, A. (2004). Matrix units and generic degrees for the Ariki-Koike algebras. Journal of Algebra, 281(2), 695-730. [More Information]
  • James, G., Mathas, A. (2004). Symmetric Group Blocks Of Small Defect. Journal of Algebra, 279(2), 566-612. [More Information]
  • Mathas, A. (2004). The Representation Theory Of The Ariki-Koike Algebras And The Cyclotomic q-Schur Algebras. Mathematical Society of Japan's 10th International Conference : Representation Theory of Algebraic Groups and Quantum Groups, Tokyo, Japan: Mathematical Society of Japan.
  • Ariki, S., Mathas, A. (2004). The representation type of Hecke algebras of type B. Advances in Mathematics, 181(1), 134-159.

2003

  • Mathas, A. (2003). Tilting modules for cyclotomic Schur algebras. Journal fuer die Reine und Angewandte Mathematik: Crelle's journal, 562, 137-169.

2002

  • James, G., Mathas, A. (2002). Equating decomposition numbers for different primes. Journal of Algebra, 258(2), 599-614.
  • Dipper, R., Mathas, A. (2002). Morita equivalences of Ariki-Koike algebras. Mathematische Zeitschrift, 240(3), 579-610. [More Information]

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