Professor Amitai Regev Department of Mathematics Ziskind Building, Room 154 Tel: +972-8-934-4210 Fax: +972-8-934-4122 amitai.regev@weizmann.ac.il

The characters of the Symmetric groups Sn — and their degrees — provide a powerful tool in the quantitative study of the polynomial identities satisfied by non commutative algebras, both associative and non associative. The description of the related asymptotic invariants was recently obtained for semi-simple Lie algebras, and also for Capelli polynomial identities.
Recently I studied various probability measures on the so called Young graph. The Vershik-Kerovs theory — and its extensions — relate these measures to the character theory on S, the infinite symmetric group. Combined with some formulas from the theory of symmetric functions, the study of these probability measures yields (infinitely) many new combinatorial and hypergeometric identities.

## Publications

1. A. Regev
• Asymptotics of Degrees of some Sn Regular Representations.
Isr. J. Math., 113 (1999) 15-28.
2. A. Regev and A. Vershik
• Asymptotics of Young Diagrams and Hook Numbers.
Electronic J. Combinatorics, 4 (1) (1997), \#R22.
3. A. Regev and D. Zeilberger
• Proof of a Conjecture of Multisets of Hook Numbers.
Ann. Combinatorics, 1 No. 4, (1997), 391--394.
4. A. Giambruno, A. Regev and M. Zaicev
• On the Codimension Growth of Finite Dimensional Lie Algebras.
J. Algebra, 220, (1999) 466-474.
5. S. Mishchenko, A. Regev and M. Zaicev
• The Eponential Growth of Codimensions for Capelli Identities.
Israel J. Math., 115 (2000) 333-342.
6. A. Regev
• $S_\infty$ Representations and Combinatorial Identities.
AMS Transactions, to appear.
7. A. Regev
• Generalized Hook and Content Number Identities.
Euro J. Comb., 21 no. 7, 949-957.
8. A. Regev
• Generalized Hook and Content Number Identities - the Projective Case.
Euro J. Comb., 21 no. 7, 959-966.
9. A. Berele and A. Regev
• Exponential Growth of some P.I. Algebras.
J. Algebra, to appear.
10. P. Cohen and A. Regev
• A Six Generalized Square Theorem, with Applications to PI Algebras.
J. Algebra, 239, 174-190 (2001).
11. G. Olshanski and A. Regev
• Random Young Tableaux and Combinatorial Identities
preprint.

Faculty of Mathematics and Computer Science
Weizmann Institute of Science
Rehovot, 76100 Israel,
Tel: 972-8-934-3540 Fax: 972-8-934-4122
Email:
math@wisdom.weizmann.ac.il