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Yang-Baxter Equation, Symmetric Functions and Grothendieck Polynomials 3 hep-th/9306005 YANG - BAXTER EQUATION, SYMMETRIC FUNCTIONS AND GROTHENDIECK POLYNOMIALS SERGEY FOMIN Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts av., Cambridge, MA 02139-4307, U.S.A. and Theory of Algorithm Laboratory SPIIRAN, St.Petersburg, Russia ANATOL N. KIRILLOV L.P.T.H.E., University Paris 6, 4 Place Jussieu, 75252 Paris Cedex 05, France and Steklov Mathematical Institute, Fontanka 27, St.Petersburg, 191011, Russia ABSTRACT New development of the theory of Grothendieck polynomials, based on an exponential solution of the Yang-Baxter equation in the algebra of projectors are given. arXiv:hep-th/9306005v2 2 Jun 1993 §1. Introduction. The Yang-Baxter algebra is the algebra generated by operators hi(x) which satisfy the following relations (cf. [B], [F], [DWA]) hi(x)hj(y) = hj(y)hi(x), if |i − j| ≥ 2, (1.1) hi(x)hi+1(x + y)hi(y) = hi+1(y)hi(x + y)hi+1(x). (1.2) The role the representations of the Yang-Baxter algebra play in the theory of quantum groups [Dr], [FRT], the theory of exactly solvable models in statistical mechanics [B], [F], [Ji], low-dimensional topology [DWA], [Jo], [RT], [KR], the theory of special functions [KR], [VK], and others branchers of mathematics (see, e.g., the survey [C]) is well-known. In this paper we continue to study (cf. [FK1], [FK2]) the connections between the Yang-Baxter algebra and the theory of symmetric functions and Schubert and Grothendieck polynomials. In fact, one can construct a family of symmetric functions 2 Sergey Fomin and Anatol N. Kirillov (in general with operator-valued coefficients) related with any integrable-by-quantum- inverse-scattering-method model [F]. This class of symmetric functions seems to be very interesting (see ,e.g., [KS]). However, in this paper we want to study “an inverse problem”, namely, what kind of “integrable models” correspond to the Schur functions [M1]. Remind that the Schur symmetric functions are the characters of irreducible, polynomial, finite dimensional representations of the Lie algebra su(n). Having in mind this “inverse problem”, let us add to the above conditions (1.1)-(1.2) an equation hi(x)hi(y) = hi(x + y), hi(0), (1.3) thus getting the so-called colored braid relations (CBR), [DWA] (see [KB], [FS], [FK1] for examples of their representations). It turns out that, once the relations (1.1)-(1.3) hold, one can introduce a whole class of symmetric functions (and even “double”, or “super” symmetric functions) and respective analogues of the (double) Schubert (Grothendieck) polynomials [L], [LS], [M2] as well. These analogues are proved to have many properties of their prototypes; e.g. we generalize the Cauchy identities and the principal specialization formula (see also [FK1]). The simplest solution of the above equations involves the nilCoxeter algebra of the symmetric group [FS]. Exploring this special case, we constructed in our previous paper [FS1] the super-analogues of Stanley’s symmetric functions Fw (see [S]), provided another combinatorial interpretation of Schubert polynomials Xw, and reproved the basic facts concerning Fw’s and Xw’s. In this paper we do the same for Grothendieck and stable Grothendieck polynomials. The Grotendieck polynomials Gw ∈ Sn, were introduced by A.Lascoux and M.-P.Schutzenberger [LS], [L] in their study of Grothendieck ring of the flag manifold. It is well-known (see e.g. [L]) that Schubert polynomials Xw corresponds to the homogeneous component of lowest degree in Grothendieck one Gw. The condition (1.2) is one of the forms of the Yang-Baxter equation (YBE); (1.3) means that we are interested in exponential solutions of the YBE. In other words, we are interested in quasitriangular R - matrices [D] with 1-dimensional auxiliary space. The natural source of such solutions is the following. Take an associative local algebra with generators ei (i.e. ei and ej commute if |i − j| ≥ 2) and define hi(x) = exp(xei). Then (1.1) and (1.3) are automatically satisfied, and one only needs the YBE (1.2). Let us sum up our main results of this paper dealing with exponential solutions of the YBE (see also [FK2]). Theorem. The following statements are equivalent i) (YBE): exp xe1 · exp(x + y)e2 · exp ye1 = exp ye2 · exp(x + y)e1 exp xe2; ii) [e1,[e1,...,[e1 ,e2] ...]]=[e2,[e2,...,[e2 ,e1] ...]] for all n. 2n 2n As a corollary we obtain the following examples of exponential solutions of the YBE: | {z } | {z } 2 2 1) e1 = de1, e2 = de2, e1e2e1 = e2e1e2 (1.4) Yang-Baxter equation, symmetric functions and Grothendieck polynomials 3 (the nilCoxeter algebra [FS] if d = 0; the algebra of projectors if d 6= 0). ′ 2) e1e2 − e2e1 = 1 (the Heisenberg salgebra) (1.5) 3) (e1,(e1,e2))=0=(e2,(e2,e1)) (the algebra U+(sl(3)) (1.6) In all these cases hi(x) = exp(xei), i =1,2. 4) (e1,(e1,e2))=0=(e2,(e2,e1)) and hi(x) = exp(x log(1 + ei)), i =1,2, (1.7) (the generalized Verma relations). The analogues of these examples for the Lie algebras of type B2 and G2 are given in Propositions 2.15 and 2.16. Insections3-5weconsiderinmoredetailthepropertiesofGrothendieckpolynomials. Among other, we propose a new development of the theory of Grothendieck polynomials based on an exponential solution (1.4) of YBE. We propose a new combinatorial rule for description of (stable) Grothendieck polynomials in terms of 0-1-matrices and give an interpretation of the Coxeter relations in the symmetric group Sn as some kind of operations (“Coxeter moves”) on the set of 0-1-matrices (cf. [BB]). We assume to consider the connections of our construction with that of A. Kohnert [Ko] in a separate publication.ItshouldbenotedthatourconstructionofGrothendieckpolynomials(based on a study of 0-1-matrices) involvs in consideration also the nonreduced decompositions of permutations. It seems a very intriguing problem to give a combinatorial/algebraic description 2 for (principal) specialization x1 = 1,x2 = 1,... (resp. x1 = 1,x2 = q,x3 = q ,...) of Grothendieck polynomials (cf. [M2], [FS]). We give in this paper only a few remarks dealing with this task (see Corollary 3.5). Acknowledgement. The authors are grateful to Alain Lascoux and Richard Stanley for helpful discussions and useful comments. §2. The Yang-Baxter equation. Let A be an associative algebra with identity 1 over a field K of zero characteristic, and let { hi(x) | x ∈ K, i =1,2,...} be a family of elements of A. We shall study the situation when hi(x)’s satisfy the following conditions: hi(x)hj(y) = hj(y)hi(x), if |i − j| ≥ 2, (2.1) hi(x)hi+1(x + y)hi(y) = hi+1(y)hi(x + y)hi+1(x), (2.2) hi(x)hi(y) = hi(x + y), hi(0) = 1. (2.3) The equation (2.2) is one of the forms of the Yang-Baxter equation (YBE) see e.g. [DWA]; (2.3) means that we are interested in exponential solutions of YBE. The most 4 Sergey Fomin and Anatol N. Kirillov natural way to construct such solutions is the following. Let e1,e2,... be generators of our algebra A, assume they satisfy locality condition eiej = ejei, if |i − j| ≥ 2. (2.4) Then let hi(x) = exp(xei), (2.5) we assume that the expression in the right-hand side is well-defined in A. Then (2.1) and (2.3) are guaranted and we only need to satisfy the YBE (2.2) which in this case can be rewritten as exp(xei) exp(xei+1) exp(yei+1) exp(yei) = = exp(yei+1) exp(yei) exp(xei) exp(xei+1), (2.6) i.e. exp(xei) exp(xei+1), exp(yei+1) exp(yei) =0. Some examples of solutions are given below. Definition 2.1. A generalized Hecke algebra Ha,b is defined to be an associative algebra with generators {ei : i =1,2,...} satisfying eiej = ej ei, |i − j| ≥ 2, eiei+1ei = ei+1eiei+1, 2 ei = aei + b. For example, H0,1 is the group algebra of the (infinite) symmetric group; H0,0 is the corresponding nilCoxeter algebra [FS]. Lemma 2.2. Let c ∈ K. The elements hi(x) ∈ Ha,b defined by ecx − 1 h (x)=1+ e (2.7) i a i satisfy (2.1)-(2.2). (This statement is implicit in [R].) ecx − 1 Proof. It is convenient to write [x] instead of . In this notation, h (x) = a i 1+[x]ei. It is easy to check that [x + y]=[x]+[y] + a[x][y] . (2.8) Now (cf. (2.2)) (1+[x]ei)(1+[x + y]ei+1)(1+[y]ei) − (1+[y]ei+1)(1+[x + y]ei)(1+[x]ei+1) = 2 2 = ([x]+[y] − [x + y])(ei − ei+1)+[x][y](ei − ei+1) = − a[x][y](ei − ei+1)+[x][y](aei + b − aei+1 − b)=0 . Yang-Baxter equation, symmetric functions and Grothendieck polynomials 5 Corollary 2.3. (Case a = 0.) Let p ∈ K. The elements hi(x) ∈ H0,b defined by hi(x)=1+ pxei (2.9) satisfy (2.1) - (2.2). Proof. Let in (2.5) c = ap and then tend a to 0. In the case b = p = 1 (the group algebra of the symmetric group) the example of the previous Corollary is well-known as the so-called Yang’s solution of the Yang-Baxter equation [Y]. Corollary 2.4. (Case b = 0.) Let c ∈ K. The elements hi(x) ∈ Ha,0 defined by (2.7) satisfy (2.1) - (2.3). Note. In fact, one can prove that in this case (2.7) can be rewritten as c h (x) = exp( xe ) .
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