Amitai Regev Touﬁk Mansour

numerative ombinatorics

A pplications Enumerative Combinatorics and Applications ECA 1:2 (2021) Interview #S3I6 ecajournal.haifa.ac.il

Interview with Amitai Regev Touﬁk Mansour

Amitai Regev is the Herman P. Taubman Professor of mathematics at the Weizmann Institute of Science. He received his doctorate from the Hebrew University of Jerusalem in 1972, under the direction of Shimshon Amitsur. Regev has made signiﬁ- cant contributions to the theory of polynomial identity rings. He developed the so-called “Regev theory” that connects polynomial identity rings to representations of the symmetric group, and hence to Young tableaux. He has made seminal contributions to the asymptotic enumeration of Young tableaux and tableaux of thick hook shape, and together with William Beck- ner proved the Macdonald-Selberg conjecture for the inﬁnite Lie algebras of type B, C, and D.

Mansour: Professor Regev, first of all, we Regev: For my Ph.D.1 research (under Amit- would like to thank you for accepting this in- sur) I chose the A⊗B problem, which was then terview. What do you think about the devel- open. At an early stage of that research, I re- opment of the relations between combinatorics alized I should determine the case G⊗G where and the rest of mathematics? G is the Grassmann algebra2. After that, plus Regev: Often there is an algebraic structure lots of work, the general case – followed. I ar- with a corresponding combinatorial structure, rive in combinatorics later in my mathematical which is induced by the corresponding algebra career, so I feel lack the overview. structure. Mansour: What was the reason you chose the Mansour: What have been some of the main Hebrew University for your Ph.D. and your ad- goals of your research? visor, Shimshon Amitsur? Regev: The goal of my research has been to Regev: At that time I thought I was mostly find and to better understand the polynomial interested in algebra. Algebra in Israel at that identities of algebras, in particular, what are time meant algebra at the Hebrew University, the polynomial identities of matrices over a and mostly - with Amitsur. field. For instance, the tensor product of two Mansour: How was the mathematics in Is- PI algebras is still a PI algebra. rael and specifically at the Hebrew University Mansour: What were your early experiences at that time? with mathematics? Regev: In ring theory, the math depart- Regev: I essentially had no math before uni- ment at the Hebrew University was one of the versity. At school (in a Kibutz) I did notice strongest in the world, and much better than that I can do most math problems easily, hence other departments in Israel. Here I would men- for university I decided to try math. tion Avinoam Mann, who substituted Amitsur Mansour: Were there specific problems that as my Ph.D. advisor during the times Amitsur made you first interested in combinatorics? was abroad. The authors: Released under the CC BY-ND license (International 4.0), Published: January 1, 2020 Toufik Mansour is a professor of mathematics at the University of Haifa, Israel. His email address is [email protected] 1See, A. Berele and Y. Roichman, The mathematics of Amitai Regev, Advances in Applied Mathematics 37 (2006) 132–138. 2See, A.P. Popov, Identities of the tensor square of the Grassmann algebra, Algebra Logic 21 (1982) 296–316 Interview with Amitai Regev

Mansour: Would you say a little bit more matrices. But I should probably include the about your most influential results and why Z2 Schur-Weyl duality (with Berele) with the they have been influential? corresponding hook Schur functions. We some- Regev: Codimensions, cocharacters and times can determine if the corresponding gen- growth were introduced in my thesis (see foot- erating is algebraic or not. For instance, mine note 1). To see their role in ring theory let result with Beckner states that when k is odd me quote from Formanek’s review of a paper number then the generating function is not al- of mine on growth: “The theory of polynomial gebraic. (3) Zeilberger’s works with Shalosh B. identities ring (PI rings) can be divided into Ekhad. two parts, a ring theoretic part, which studies Mansour: What are the top three open ques- properties of PI rings, and a quantitative part tions in your list? which studies the T -ideal of identities. Until Regev: (1) Determine the exact asymptotics 1 about 1985, the greater part of research in PI cn(Mk(F )) (see footnote ). Very recently, in a rings was of the former type, but since then joint work with Berele, we showed a fairly tight the second is predominated.” Growth was in- sandwich theorem for those invariants. With b n troduced and studied in my thesis and was ex- the notations cn(Mk(F )) ∼ an r it follows tended by several authors. that b is half-integer, r is a positive integer The Schur-Weyl duality is of fundamental according to a result of Giambruno– Zaicev, importance in all areas of math, in particu- however, there is still no information about lar in combinatorics. The asymptotics of cor- the constant a. Thus, it is interesting to de- responding combinatorial sums is essential in termine the constants a and b. (2) In a joint that study. With my student Allan Berele we work with Beckner, we proved that cn(Mk(F )) developed a Z2-analog of the Schur-Weyl du- is non-algebraic if k is odd. My previously ality. It had applications to (new) symmetric mentioned result with Beckner solves partially functions, as well as to the theory of polyno- a problem of R. Stanley. The corresponding mial identities. question when k is even - is open. The obvious Sets of integer partitions correspond to sets conjecture is: this is true for all k; no doubt it of Young tableaux, and these give rise to the is true. (3) Similar works for cn(Mk(G)) (see corresponding combinatorial sums – with their footnote 1) where G is the Grassmann algebra. corresponding asymptotics. Also, the asymp- Also, to determine a, b and r for the verbally totics of the corresponding combinatorial sum prime algebras. imply interesting relations with the celebrated Mansour: What kind of mathematics would Mehta-Selberg integral. Surprisingly, that fun- you like to see in the next ten-to-twenty years damental integral allows the exact computa- as the continuation of your work? tion of the asymptotics of the polynomial iden- Regev: I would prefer a more concrete and tities of matrices. less so-called “abstract nonsense”. Also, I Mansour: What would guide you in your re- would like to see more results of the collab- search? A general theoretical question or a oration of Zeilberger and Shalosh B. Ekhad3. specific problem? Mansour: What would you say about some of Regev: Usually, it would be a specific prob- the major directions in combinatorics for the lem. next two decades? Mansour: What three results related to your Regev: The combinatorics of superalgebras4. work do you consider the most influential in Mansour: Do you think that there are core or combinatorics during the last thirty years? mainstream areas in mathematics? Are some Regev: Two of my own research(see footnote topics more important than others? 1) and one of Zeilberger: (1) A ⊗ B theorem Regev: No doubt certain areas of mathemat- introducing “growth”. (2) The Mehta-Selberg ics have more prestige than others. Number integral and the fact that it allows us com- theory is more appealing than most other puting the asymptotics of the codimension se- areas. quence cn(A), where A is the algebra of k × k Mansour: What do you think about the dis- 3See, https://sites.math.rutgers.edu/ zeilberg/pj.html 4V.G. Kac, C. Martinez and E. Zelmanov, Graded simple Jordan superalgebras of growth one, Memoirs of the AMS Series 711, 2001.

ECA 1:2 (2021) Interview #S3I6 2 Interview with Amitai Regev tinction between pure and applied mathemat- mutations. What do you think about the re- ics that some people focus on? Is it meaningful search in this direction? at all in your own case? How do you see the Regev: In my work, I somehow avoided count- relationship between so-called “pure” and “ap- ing Pattern-Avoiding permutations [“The plied” mathematics? Stanley-Wilf” Conjecture]. A few years ago Regev: Such a distinction probably exists, but I asked Wilf why they made that conjecture? the intersection of pure and applied mathemat- He said: “Because of your results”. ics is large. For example, the character theory Mansour: Would you tell us about your of Sn can later be considered pure, but it is thought process for the proof of one of your applied in physics. most important results? How did you become Mansour: What advice would you give to interested in that problem? How long did it young people thinking about pursuing a re- take you to figure out a proof? Did you have search career in mathematics? a “eureka moment”? Regev: Try to find a topic with interesting Regev: Eureka moment occurred for the A⊗B and/or natural applications. It seems that peo- problem (see footnote 1,3) when I realized that ple are getting tired from too abstract and ir- even the G ⊗ G case is still open. relevant applications. Nevertheless, work hard Mansour: Is there a specific problem you and do not give up. have been working on for many years? What Mansour: Would you tell us about your in- progress have you made? terests besides mathematics? Regev: Yes, several such problems. For ex- Regev: Playing Baroque music on the Oboe, ample, (1) with Alan Berele, the asymptotics with a small group of Baroque players. But of the cocharacters of the ring of matrices over right now I am not healthy enough for that. a field (see footnote 1). At the end, this was Mansour: Before we close this interview with determined by an application of the Mehta- one of the foremost experts in combinatorics, Selberg integral. (2) Various statistics on Sn we would like to ask some more specific math- were introduced by MacMahon who studied ematical questions. The applications of the their equidistributions, where with Roichman representation theory of the symmetric group we extended these theorems to correspond- to the study of algebras satisfying polynomial ing q-analogues. (3) The existence of cen- identities played an important role in your re- tral polynomials has important consequences search. You used Young diagrams many times for the structure of the algebra. Kaplansky and obtained a series of important results in asked if central polynomials on matrices exist. this field. Would you describe the main ideas Then Formanek and, indepenedently, Razmis- and the combinatorial methods you used in lov proved: Yes, they exist. For k × k ma- this direction? trices, I constructed such central polynomials Regev: Given n and a PI algebra A, we mea- as follows: split k2 xs into k blocks of xs of sure how large is the part of non-identities rel- lengths 1, 3,... and similarly split k2 ys. Now ative to the whole space. A significant simpli- shuffle these blocks, and finally alternate the fication is obtained when restricting to multi- xs and independently the ys. The result is a linear identities. This brings in the represen- polynomial Pk[x1, . . . , xk2 , y1, . . . , yk2 ] which I tations of Sn. This yields the sequence of codi- introduced nearly 40 years ago. It is easy to mensions of A. Similarly for the corresponding see that Pk[x, y] is central (i.e. takes central sequence representation of Sn. These two se- values on k × k matrices). I checked that these quences determine a lot about the PI algebra polynomials are non-zero for k = 2 and k = 3. A, hence the relations to Young diagrams. Then I conjectured that Pk[x, y] is non-zero for Mansour: The study of permutation patterns all k. At the time it seemed an extremely has seen great advances in the last thirty years. difficult problem. Then Formanek found an One of the earliest non-trivial result was ob- incredibly ingenious proof which proved that tained by you that gives the growth rate of the conjecture. These central polynomials Pk[x, y] number of the permutations avoiding a mono- already found several applications. (Some peo- tone pattern of length k. But we still do not ple call Pk[x, y] “the Formanek-Regev” central know how to enumerate the 1324-avoiding per- polynomials in PI Theory.)

ECA 1:2 (2021) Interview #S3I6 3 Interview with Amitai Regev

Mansour: Professor Amitai Regev, I would terview on behalf of the journal Enumerative like to thank you for this very interesting in- Combinatorics and Applications.

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