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The Yang-Baxter Equation, Symmetric Functions, and Schubert Polynomials

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The Yang-Baxter Equation, Symmetric Functions, and Schubert Polynomials DISCRETE MATHEMATICS ELSEVIER Discrete Mathematics 153 (1996) 123 143 The Yang-Baxter equation, symmetric functions, and Schubert polynomials Sergey Fomin a'b'*, Anatol N. Kirillov c a Department of Mathematics, Room 2-363B, Massachusetts Institute of Technolooy, Cambridge, MA 02139, USA b Theory of Aloorithms Laboratory, SPIIRAN, Russia c Steklov Mathematical Institute, St. Petersbur9, Russia Received 31 August 1993; revised 18 February 1995 Abstract We present an approach to the theory of Schubert polynomials, corresponding symmetric functions, and their generalizations that is based on exponential solutions of the Yang-Baxter equation. In the case of the solution related to the nilCoxeter algebra of the symmetric group, we recover the Schubert polynomials of Lascoux and Schiitzenberger, and provide simplified proofs of their basic properties, along with various generalizations thereof. Our techniques make use of an explicit combinatorial interpretation of these polynomials in terms of configurations of labelled pseudo-lines. Keywords: Yang-Baxter equation; Schubert polynomials; Symmetric functions 1. Introduction The Yang-Baxter operators hi(x) satisfy the following relations (cf. [1,7]): hi(x)hj(y) = hj(y)hi(x) if li -j[ >_-2; hi(x)hi+l (x + y)hi(y) = hi+l (y)hi(x + y)hi+l(X); The role the representations of the Yang-Baxter algebra play in the theory of quantum groups [9], the theory of exactly solvable models in statistical mechanics [1], low- dimensional topology [7,27,16], the theory of special functions, and other branches of mathematics (see, e.g., the survey [5]) is well known. * Corresponding author. 0012-365X/96/$15.00 (~) 1996--Elsevier Science B.V. All rights reserved SSD10012-365X(95)O0132-8 124 S. Fomin, A.N. Kirillov/Discrete Mathematics 153 (1996) 123 143 We study the connections between the Yang-Baxter algebra and the theory of sym- metric functions and Schubert polynomials. Let us add to the above conditions the equation hi(x)hi(y) ~- hi(x --}- y), thus getting the so-called colored braid relations (see [17, 14] for examples of their representations). It turns out that, once these relations hold, one can introduce a whole class of symmetric functions (and even 'double', or 'super-' symmetric functions) and respective analogues of the [double] Schubert polynomials [22,25] as well. These analogues are proved to have many of the properties of their prototypes; e.g., we generalize the Cauchy identities and the principal specialization formula. The simplest solution of the above equations involves the nilCoxeter algebra of the symmetric group [14]. Exploring this special case, we construct super-analogues of Stanley's symmetric functions Gw (see [29]), provide another combinatorial interpre- tation of Schubert polynomials ~w of Lascoux and Schiitzenberger, and reprove the basic facts concerning Gw's and ~w'S. Recently, the construction of this paper has been used [2] to produce a Pieri rule for Schubert polynomials and yet another algorithm that generates the monomials of ~w. Other solutions of the main relations are also given. One of them involves Hecke algebras, another one the universal enveloping algebra of the Lie algebra of nilpotent upper triangular matrices. In this paper, we intended to emphasize the power of the 'geometric approach' (Sections 3-4) that allows to derive algebraic identities about hi(x)'s by modifying, according to certain rules, the corresponding configurations of labelled pseudo-lines. This is why some of our proofs appear to look like just 'See Fig. X' (cf. proofs of Proposition 6.4, Theorem 8.1(i), etc.). 2. The Yang-Baxter equation Let d be an associative algebra with identity 1 over a field K of zero characteristic, and let {hi(x) : x E K, i = 1,2 .... } be a family of elements of d. (In fact, we will treat x as a formal variable rather than a parameter.) We shall study situations where hi(x)'s satisfy the following conditions: hi(x)hj(y) = hj(y)hi(x) if [i -j] t>2; (2.1) hi(x)hi+l(x + y)hi(y) = hi+l(y)hi(x + y)hi+l(x); (2.2) hi(x)hi(y) = hi(x + y); hi(O ) = 1. (2.3) The condition (2.2) is one of the forms of the Yang-Baxter equation (YBE); (2.3) means that we are interested in exponential solutions of the YBE. The most natural S. Fomin, A.N. Kirillov / Discrete Mathematics 153 (1996) 123 143 125 way to construct such solutions is the following. Let Ul, u2, • . be generators of our algebra d; assume they satisfy uiuj = ujui, li - Jl >/2; (2.4) i,e., ,~¢ is a local alyebra in the sense of [30]. Then let hi(x) = exp(xui); (2.5) we assume that the expression on the right-hand side is well-defined. Then (2.1) and (2.3) are guaranteed and we only need to satisfy the YBE (2.2) which in this case can be rewritten as exp(xui) exp((x + y)ui+l ) exp(yui) = exp(yui+l ) exp((x + y)ui) exp(xui+l ). (2.6) Some examples of solutions are given below. Definition 2.1. A [generalized] Hecke algebra (sometimes also called an Iwahori algebra) .~,b is an associative algebra with generators {ui:i = 1,2,...} satisfying (2.4), UiUi+l Ui ~ Ui+l Ui~li+l, (2.7) and u 2 = aui 4- b. (2.8) In particular, ~0,1 is the group algebra of the symmetric group. The corresponding nilCoxeter alyebra ~o, o (see [14]) defined by (2.4), (2.7), and u 2 = 0 can be interpreted as the algebra spanned by permutations of Sn, with the multiplication rule usual product wv if l(w) + l(v) = l(wv), w - v = 0 otherwise, where l(w) is the length of a permutation w (the number of inversions). It is not hard to cheek that (2.6) holds in ~a, b if b = 0. However, we will give an indirect proof of this fact, in order to relate it to some well-known properties of Hecke algebras. The following statement is implicit in [28]. Lemma 2.2. Let c E K, The elements hi(x) E ~, b defined by e CX - 1 hi(x) = 1 + Ui (2.9) a satisfy (2.1)-(2.2). 126 S. Fomin, A.N. Kirillov/Discrete Mathematics 153 (1996) 123-143 Proof. It is convenient to write [x] instead of (e cx - 1)/a. In this notation, hi(x ) ~- 1 + [x]ui. It is easy to check that [x + y] = [x] + [y] + a[x][y]. Now (cf. (2.2)) (1 + [X]Ui)(1 + [X + y]ui+l)(1 q-[y]ui) - (1 + [y]ui+l)(1 + Ix + y]ui)(1 + [x]ui+l) --= (Ix] + [y] - [x + y])(ui - Ui+l) + [x][y](u~ -- U~+I) = -- a[x][y](ui -- ui+ 1 ) + [x][y](aui + b - aui+l - b) = O. [] Corollary 2.3 (case a = 0). The elements hi(x) E 0¢t°o, b defined by hi(x) = 1 + xui satisfy (2.1)-(2.2). Proof. In (2.9), let c = a and then tend a to 0. [] In the case a = 0, b = 1 (the group algebra of the symmetric group) the example of the previous corollary is well-known as the so-called Yang's solution [31] of the Yang-Baxter equation. Corollary 2.4 (case b = 0). Let c E K. The elements hi(x) E ~Yga, O defined by (2.9) satisfy (2.1)-(2.3). Proof. In this case (2.9) can be rewritten as hi(x) = exp(Cxui), and (2.3) follows. [] In particular, (2.1)-(2.3) hold in the case a = b = 0 [14, Lemma 3.1]. Thus the elements hi(x) = 1 + xui of the nilCoxeter algebra of the symmetric group provide an exponential solution of the Yang-Baxter equation. (This can also be easily checked directly. ) 3. Geometric interpretation The relations (2.1)-(2.2) are known to have a nice geometric interpretation (see, e.g., [6]) which is reproduced below; in the next section this interpretation will be modified to involve the condition (2.3) as well. Suppose we have a family of non-vertical straight lines intersecting a vertical strip on a real plane; no three of these lines meet at the same point. Also assume that an indeterminate is associated with each line. A typical example is presented in Fig. 1. Given such a configuration with n lines, one can define a sequence Sa, " "" sa,, of adjacent transpositions (a reduced decomposition in the symmetric group Sn) as shown on Fig. 1; in other words, the index ai of each sa, indicates which two of adjacent lines (counting bottom-up) get interchanged when we pass the ith intersection point (counting from the left). The product of these generators in the symmetric group corresponds to the permutation defined by a given configuration. S. Fomin. A.N. Kirillov/Discrete Mathematics 153 (1996) 123-143 127 Ll X L3 L 2 X X 4"-,. Sl S 3 S 2 Sl S 3 Fig. 1. Assume conditions (2.1)-(2.2) are satisfied by some elements {hi(x)}. Let ~ be a configuration of the above-described type. Define ~((~; X1, X2,...) = ha, (xkt - xt, )ha2 (xk: - xl2 )"" hap (xk,, - xtp ), (3.1) where, as before, (al ..... ap) is a reduced decomposition corresponding to the given configuration, and xk, and xt, are the indeterminates for the lines meeting at the ith intersection point; xk, corresponds to a line with a smaller slope and xt, to a line with a greater slope. For example, if ~ is the configuration in Fig. 1, then ~(~;XI,X2,X3,X4) ~- hi(x2 -Xl)h3(x4 -x3)h2(x4 -xl)hl(x4 -x2)h3(x3 -Xl).
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