Andrew Mathas

I am an associate professor in the School of Mathematics and Statistics at the University of Sydney.

Andrew Mathas School of Mathematics and Statistics F07 University of Sydney, NSW 2006. Australia Office 635 Carslaw +61 2 9351 6058 (W) +61 2 9351 4534 (Fax) mathas@maths.usyd.edu.au

Research Interests

My main area of interest and expertise is the representation theory of Coxeter groups, Iwahori-Hecke algebras and Schur algebras. Recently my work has focused on the modular representation theory of the Ariki-Koike algebras and the associated cyclotomic q-Schur algebras. 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:

• 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 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.

I am a member of the Algebra research group.

Publications

Alex and Erika (joint with M.)

Preprints

• Cyclotomic Solomon Algebras, Adv. Math., to appear. With Rosa Orellana.
• Morita equivalences of the cyclotomic Hecke algebras of type, G(r,p,n), J. Reine Angew. Math., to appear. With Jun Hu
• Seminormal forms and Gram determinants for cellular algebras, J. Reine Angew. Math., 619 (2008), to appear. With an appendix by Marcos Soriano

Book

• Iwahori-Hecke algebras and Schur algebras of the symmetric group, University lecture series, 15, AMS, 1999.                            errata

Papers

• Blocks of cyclotomic Hecke algebras, Adv. Math., 216 (2007), 854-878. With Sinéad Lyle.
• Cyclotomic Nazarov-Wenzl algebras, Nagoya Math. J., 182 (2006), 47-134. With Susumu Ariki and Hebing Rui. (Special issue in honour of George Lusztig.)
• Rouquier blocks, Math. Z., 252 (2006), 511-531. With Gordon James and Sinéad Lyle.
• Row and column removal theorems for homomorphisms of Specht modules and Weyl modules, J. Alg. Comb., 22 (2005), 151-179. With Sinéad Lyle.
• Elementary divisors of Specht modules, European J. Combinatorics, 26 (2005), 943-964. With Matthias Künzer. (Special issue showcasing algebraic combinatorics.)
• Matrix units and generic degrees for the Ariki-Koike algebras, J. Algebra, 281 (2004), 695--730.
• Symmetric group blocks of small defect, J. Algebra, 279 (2004), 566--612. With Gordon James,
• Hecke algebras with a finite number of indecomposable modules, Representation theory of algebraic groups and quantum groups, Adv. Studies Pure Math., 40 (2004), 17-25. With Susumu Ariki.
• The representation theory of the Ariki-Koike and cyclotomic q-Schur algebras, Representation theory of algebraic groups and quantum groups, Adv. Studies Pure Math., 40 (2004), 261--320.
• The representation type of Hecke algebras of type B, Adv. Math., 181 (2004), 134-159. With Susumu Ariki.
• Tilting modules for cyclotomic Schur algebras, J. Reine Angew. Math., 562 (2003), 137-169.
• Equating decomposition numbers for different primes, J. Algebra, 258 (2002), 599-614. With Gordon James.
• Morita equivalences of Ariki-Koike algebras, Math. Z., 240 (2002), 579-610. With Richard Dipper,
• The Jantzen sum formula for cyclotomic q-Schur algebras, Trans. AMS, 352 (2000), 5381-5404. With Gordon James.
• The number of simple modules of the Hecke algebras of type G(r,1,n), Math. Zeitschrift, 233 (2000), 601-623. With Susumu Ariki.
• The irreducible Specht modules in characteristic 2, Bull. Lond. Math. Soc., 31 (1999), 457-462. With Gordon James.
• The Murphy operators and the centre of the Iwahori-Hecke algebras of type A, J. Alg. Comb., 9 (1999), 295-313.
• Cyclotomic q-Schur algebras, Math. Zeitschrift, 229 (1998), 385-416. With Richard Dipper and Gordon James.
• Symmetric cyclotomic Hecke algebras, J. Algebra, 205 (1998), 275-293. With Gunter Malle.
• The (Q,q)-Schur Algebra, Proc. Lond. Math. Soc., 77 (1998), 327-361. With Richard Dipper and Gordon James.
• Simple modules of Ariki-Koike algebras, Proc. Pure Symp. Math., 63 (1998), 383-396.
• A q-analogue of the Jantzen-Schaper theorem, Proc. Lond. Math. Soc., 74 (1997), 241-274. With Gordon James.
• Hecke algebras of type A at q=-1, J. Algebra, 184 (1996), 102-158. With Gordon James.
• On the left cell representations of Iwahori-Hecke algebras of finite Coxeter groups, J. London Math. Soc., 54 (1996), 475-488.
• Some generic representations, W-graphs, and duality. J. Algebra, 170 (1994), 322-353.
• A q-analogue of the Coxeter complex. J. Algebra, 164 (1994), 831-848.

Gap packages

• Specht A Gap package for calculating decomposition matrices of Hecke algebras of type A.
• Murphy These programs implement the Murphy basis of the Iwahori-Hecke algebra of the symmetric group using Chevie, version 3.4.

Some interesting links (mostly) in Sydney

Salutary wisdom from the Cycle Messengers' Guidebook to San Francisco.
1. At night, you're much safer on a bike than on foot or on public transport.
2. If you're in a neighbourhood that seems dangerous, it probably is.
3. Don't buy any drugs on the street, you'll get ripped off.
4. Obey all traffic laws when in the presence of a motorcycle cop.
5. Keep one eye out for car doors, one eye out for potholes, one eye out for pedestrians and one eye out for vehicular traffic (better get some more eyes!).
6. Cars blow through red lights all the time. Don't trust traffic lights.
7. If you're gonna take on a car driver, be prepared to fight. The automobile reigns supreme in the eyes of the feeble minded and hand guns are abundant.

Democrat Senator John Cherry had a variation on the lightbulb joke for a QUT students' forum:

How many Liberal education ministers does it take to change a lightbulb?

Three:

• Amanda Vanstone to cut funding and force the lights off.
  
• David Kemp to sneak in under the cloak of darkness, flog all the lightbulbs and privatise them;
  
• Brendan Nelson to set up an inquiry to retrospectively justify why students are better off being kept in the dark.

If I were a Springer-Verlag Graduate Text in Mathematics, I would be J.-P. Serre's Linear Representations of Finite Groups. My creator is a Professor at the College de France. He has previously published a number of books, including Groupes Algebriques et Corps de Classes, Corps Locaux, and Cours d'Arithmetique (A Course in Arithmetic, published by Springer-Verlag as Vol. 7 in the Graduate Texts in Mathematics).

Which Springer GTM would you be? The Springer GTM Test