Irreducible representations of the plactic monoid of rank four

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1 Lukasz Kubat (joint work with Ferran Cedó and Jan Okniński) Institute of Mathematics University of Warsaw Groups, Rings and the Yang Baxter equation Spa, June 23, 2017

2 Plan of the talk 1 Background Plactic monoids Motivations 2 Irreducible representations of plactic monoids Basic properties of plactic algebras Irreducible representations of the plactic monoid of rank 3 Irreducible representations of the plactic monoid of rank 4 3 Open problems

3 Plactic monoids For an integer n 1 the finitely presented monoid M n defined by generators a 1,..., a n and (Knuth s) relations a ia k a j = a k a ia j for i j < k, a ja ia k = a ja k a i is called the plactic monoid of rank n. for i < j k If K is a field then K[M n] denotes the corresponding monoid algebra, called the plactic algebra of rank n. The origins of the plactic monoid come from the work of Schensted and Knuth concerning certain combinatorial problems and operations on Young tableaux.

4 The elements of M n can be uniquely written in a canonical form that allows to identify them with semistandard Young tableaux. This leads to very deep applications of the plactic monoid in several areas of mathematics (cf. Fulton s monograph [1]) including: representation theory of the full linear group and the theory of symmetric functions, via the Littlewood Richardson rule, quantum groups (in the context of crystal bases), and the area of classical Lie algebras. According to Schützenberger, the plactic monoid should be considered among the most fundamental monoids in algebra. Despite the fact that the plactic monoid has already become an algebraic folklore, it still attracts attention of mathematicians and both the plactic monoid and its modifications are the object of intense research in last years.

5 Motivations In view of the role played by the plactic monoid, the search for irreducible representations of M n seems a natural task. Moreover, such representations may lead to completely new information about this monoid (e.g. irreducible representations of M 3 turned out to be a key tool in proving that M 3 does satisfy a nontrivial semigroup identity). On the other hand, our motivations also come from recently obtained results for the related class of the so-called Chinese monoids. Namely, the Chinese monoid C n of rank n 1 is defined by generators a 1,..., a n and relations It is known (cf. [4]) that: a ja k a i = a k a ja i = a k a ia j for i j k. the Jacobson radical J(K[C n]) of the corresponding monoid algebra K[C n] over a field K is a nilpotent ideal. if K is algebraically closed and nondenumerable then all irreducible representations of C n (simple K[C n]-modules) have a remarkably simple form and they can be built inductively from irreducible representations of the monoid C 2 (which are very easy to determine).

6 Basic properties of plactic algebras For the plactic algebra K[M n] it was previously proved (cf. [3]) that: K[M n] has finite Gelfand Kirillov dimension, but it is neither noetherian nor PI (in case n 2). if n 2 then K[M n] is prime and semiprimitive (J(K[M n]) = 0). K[M 3] has exactly two minimal prime ideals P 1 and P 2. Each of them is principal and of the form P i = Span K {x y : (x, y) ρ i} for some homogeneous congruence ρ i on M 3 (in such case we say that P i is determined by a congruence on M 3). We also have J(K[M 3]) = 0. if n 4 then K[M n] is not even semiprime! Moreover, the algebra K[M 4] has a minimal prime ideal which is not determined by a congruence on M 4. The last result suggests that the structure (and representations) of the plactic algebra of rank n 4 should be much more complex than in case n 3.

7 Irreducible representations of plactic monoids of rank 3 Recall that a (linear) representation M End K(V ) of a monoid M is said to be monomial, if there exists a basis B of V such that for every x M and every v B there exist λ K and w B satisfying xv = λw. Theorem If K is an algebraically closed and nondenumerable field then all irreducible representations of the plactic algebra K[M n] of rank n 3 are monomial. In fact, much more is known. Namely, the paper [3] contains: a full description of all simple left and right K[M 3]-modules together with corresponding primitive ideals. a proof that minimal prime ideals P 1 and P 2 of K[M 3] are semiprimitive by presenting them as intersections of annihilators of simple modules constructed in this paper. Hence J(K[M 3]) P 1 P 2. a proof that P 1 P 2 = 0, which implies that the algebra K[M 3] is semiprimitive.

8 Irreducible representations of the plactic monoid of rank 4 The starting point: using some equalities in M 4 it is possible to construct ideals I 1, I 2, I 3, I 4 of K[M 4] (of a very special type: they are finitely generated by elements of the form x y, where x, y M 4 are of equal length with respect to each of a 1, a 2, a 3, a 4) for which the ideal I 1 I 2 I 3 I 4 is nilpotent. This implies that each simple K[M 4]-module contains at least one of the ideals I 1, I 2, I 3, I 4 in its annihilator. Proposition Let V be a K-vector space with basis {e ijk : i, j, k 0}. Let α, β, γ K and γ 0. Then the action of generators a 1, a 2, a 3, a M 4 on V defined by 4 { a 1e ijk = e i+1,j,k, a 2e ijk = e i 1,j+1,k if i > 0, αe 0,j,k if i = 0, { { a 3e ijk = e i,j 1,k+1 if j > 0, γe i,j,k 1 if k > 0, a 4e ijk = βe i,0,k if j = 0, 0 if k = 0 makes V a simple K[M 4]-module with annihilator containing I 1. Moreover, it turns out that I 1 is equal to the intersection of annihilators of constructed simple modules. Hence I 1 is semiprimitive and J(K[M 4]) I 1.

9 Then, using some antiendomorphisms of the algebra K[M 4] and of its homomorphic images, it is possible to derive semiprimitivity of the ideal I 2 from semiprimitivity of I 1. Hence J(K[M 4]) I 2 as well. Next step is to construct families of simple K[M 4]-modules containing I 3 or I 4 in their annihilators. One of such families (presented below) leads to irreducible representations of M 4 which are not monomial! Proposition Let V be a K-vector space with basis {e ijk : i, j, k 0}. Let 0 α, β, γ K. Then the action of generators a 1, a 2, a 3, a 4 M 4 on V defined by a 1e ijk = e i+1,j,k, { e i 1,j+1,k if i > 0, a 2e ijk = 0 if i = 0, e i,j 1,k+1 if j > 0, a 3e ijk = βe i,0,k αβγ 1 e i 1,1,k + αγ 1 e i 1,0,k+1 if j = 0 and i > 0, e 0,0,k if j = 0 and i = 0, { γe i,j,k 1 if k > 0, a 4e ijk = 0 if k = 0 makes V a simple K[M 4]-module with annihilator containing I 3.

10 Although we do not know whether the ideals I 3 and I 4 are semiprimitive, it can be shown that some power of J(K[M 4]) is contained in I 3 I 4. Since J(K[M 4]) I 1 I 2, we conclude that some power of J(K[M 4]) is contained in I 1 I 2 I 3 I 4. Hence we get Theorem The Jacobson radical J(K[M 4]) of the plactic algebra of rank four is nilpotent. Moreover, if ρ I = {(x, y) M 4 M 4 : x y I} denotes the congruence on M 4 determined by an ideal I of K[M 4], then one can show that ρ J(K[M4 ]) = ρ Ann(V ), where V runs through all familes of constructed simple K[M 4]-modules. In particular, the monoid M 4/ρ J(K[M4 ]) is a subdirect product of the images of M 4 in the corresponding endomorphism algebras. Details (e.g. proofs and explicit form of constructed simple modules) can be found in [2] (the paper is quite long, 39 pages).

11 Open problems Due to the important role played by the plactic monoid in algebra and its applications, it seems natural to search for solution of the following (perhaps very difficult in full generality) Problem Determine all irreducible representations (at least over an algebraically closed and nondenumerable field) of the plactic monoid M n of any finite rank n. On the other hand, related important questions are concerned with the radical of the plactic algebra. They are also motivated by general (very difficult) open problems on the Jacobson radical of finitely presented algebras. Conjecture The Jacobson radical J(K[M n]) of the plactic algebra K[M n] of any finite rank n over an arbitrary field K is a nilpotent ideal.

12 Bibliography W. Fulton Young Tableaux, London Math. Soc. Stud. Texts, vol.35, Cambridge Univ. Press, New York, F. Cedó, L. Kubat, J. Okniński Irreducible representations of the plactic algebra of rank four, J. Algebra (2016), L. Kubat, J. Okniński Plactic algebra of rank 3, Semigroup Forum 84 (2012), L. Kubat, J. Okniński Irreducible representations of the Chinese monoid, J. Algebra 466 (2016), Thank you!

arxiv: v1 [math.rt] 4 Jan 2016

arxiv: v1 [math.rt] 4 Jan 2016 IRREDUCIBLE REPRESENTATIONS OF THE CHINESE MONOID LUKASZ KUBAT AND JAN OKNIŃSKI Abstract. All irreducible representations of the Chinese monoid C n, of any rank n, over a nondenumerable algebraically closed