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Symmetric Functions 7 Symmetric Functions 7.1 Symmetric Functions in General The theory of symmetric functions has many applications to enumerative combi- natorics, as well as to such other branches of mathematics as group theory, Lie algebras, and algebraic geometry. Our aim in this chapter is to develop the basic combinatorial properties of symmetric functions; the connections with algebra will only be hinted at in Sections 7.18 and 7.24, Appendix 2, and in some exercises. Let x = (x\, Jt2,...) be a set of indeterminates, and let n eN. A homogeneous symmetric function of degree n over a commutative ring R (with identity) is a formal power series where (a) a ranges over all weak compositions a = («i, «2, • • •) of n (of infinite a x length), (b) ca e R, (c) x stands for the monomial x\ x£ ..., and (d) /(x^i), Xw(2), • • •) = f(x\,X2,.. •) for every permutation w of the positive integers P. (A symmetric function of degree 0 is just an element of R.) Note that the term "symmetric function" is something of a misnomer; f(x) is not regarded as a function but rather as a formal power series. Nevertheless, for historical reasons we adhere to the above terminology. The set of all homogeneous symmetric functions of degree n over R is denoted n n A R. Clearly if /, g e A R and a, b e R, then af + bg e A\\ in other words, A^ is an R-module. For our purposes it will suffice to take R = Q (or sometimes Q with some indeterminates adjoined), so A^ is a Q-vector space. For the sake of convenience, then, and because some readers are doubtless more comfortable with vector spaces than with modules, we will henceforth work over Q, though this is not the most general approach. If / € AQ and g e AQ, then it is clear that fg € AQ+W (where fg is a product of formal power series). Hence if we define AQ = AQ 0 AQ 0 • • • (vector space direct sum) (7.1) 286 Downloaded from https://www.cambridge.org/core. Access paid by the UC Berkeley Library, on 26 Jun 2019 at 21:59:31, subject to the Cambridge Core terms of use, availableCambridge at https://www.cambridge.org/core/terms Books Online © Cambridge University. https://doi.org/10.1017/CBO9780511609589.006 Press, 2010 7.2 Partitions and Their Orderings 287 (so the elements of AQ are power series / = /o+/H— where /„ e AQ and all but finitely many fn = 0), then AQ has the structure of a Q-algebra (i.e., a ring whose operations are compatible with the vector space structure), called the algebra (over Q) of symmetric functions. Note that the algebra AQ is commutative and has an identity element 1 e AQ. The decomposition (7.1) in fact gives AQ the structure of a graded algebra, meaning that if / e AQ and g e AQ, then fg e AQ+n . From rt now on we suppress the subscript Q and write simply A and A for AQ and AQ. Note, however, that in the outside literature A usually denotes Az. A central theme in the theory of symmetric functions is to describe various bases of the vector space A" and the transition matrices between pairs of these bases. We will begin with four "simple" bases. In Sections 7.10-7.19 we consider a less obvious basis which is crucial for the deeper parts of the theory. 7.2 Partitions and Their Orderings Recall from Section 1.3 that a partition A of a nonnegative integer n is a sequence h n (Ai,..., Xk) e N satisfying Ai > • • • > Xk and J2 h — - Any A,; = 0 is consi- dered irrelevant, and we identify A with the infinite sequence (Ai,..., A*, 0,0,...). We let Par(n) denote the set of all partitions of n, with Par(0) consisting of the empty partition 0 (or the sequence (0,0,...)), and we let Par:= For instance (writing for example 4211 as short for (4, 2, 1, 1, 0,...)), Par(l) = {1} Par(2) = {2, 11} Par(3) = {3,21,111} Par(4) = {4,31,22,211,1111} Par(5) = {5,41,32,311,221,2111, 11111}. If A e Par(/i), then we also write A h n or |A| = n. The number of parts of A (i.e., the number of nonzero A/) is the length of A, denoted £(A). Write m, = m;(A) for the number of parts of A that equal /, so in the notation of Section 1.3 we have A = (lmi2m2...), which we sometimes abbreviate as lmi2m2 • • •. Also recall from the discussion of entry 10 of the Twelvefold Way in Section 1.4 that the conjugate r partition X' = (A p A^,...) of A is defined by the condition that the Young (or Ferrers) diagram of A; is the transpose of the Young diagram of A; equivalently, f m,-(A/) = A, - A?+i. Note that X\ = l(X) and Ai = l(X ). Three partial orderings on partitions play an important role in the theory of symmetric functions. We first define /x c A for any /x, A e Par if /z; < A/ for all /. If we identify a partition with its (Young) diagram, then the partial order Downloaded from https://www.cambridge.org/core. Access paid by the UC Berkeley Library, on 26 Jun 2019 at 21:59:31, subject to the Cambridge Core terms of use, availableCambridge at https://www.cambridge.org/core/terms Books Online © Cambridge University. https://doi.org/10.1017/CBO9780511609589.006 Press, 2010 288 7 Symmetric Functions Figure 7-1. Young's lattice. c is given simply by containment of diagrams. The set Par with the partial order c is called Young's lattice Y and (as mentioned in Exercise 3.63) is isomorphic to Jf(N2). The rank of a partition A in Young's lattice is equal to the sum |A| of its parts, so the rank-generating function by equation (1.30) is given by See Figure 7-1 for the first six levels of Young's lattice. The second partial order is defined only on Par(«) for each n eN, and is called dominance order (also known as majorization order or the natural order), denoted <. Namely, if |/x| = |A.| then define /z < A if + A&2 H h fit < A.i + A2 H h A/ for all i > 1. For the reader's benefit we state the following basic facts about dominance order, though we have no need for them here. • (Par(n), <) is a lattice. • The map X \-+ A' is an anti-automorphism of (Par(jz), <) (so Par(n) is self-dual under dominance order). • (Par(rc), <) is a chain if and only if n < 5. • (Par(?z), <) is graded if and only if n < 6. Downloaded from https://www.cambridge.org/core. Access paid by the UC Berkeley Library, on 26 Jun 2019 at 21:59:31, subject to the Cambridge Core terms of use, availableCambridge at https://www.cambridge.org/core/terms Books Online © Cambridge University. https://doi.org/10.1017/CBO9780511609589.006 Press, 2010 7.3 Monomial Symmetric Functions 289 n • For k = (k\,..., kn) h n, \ctVx denote the convex hull in R of all permutations of the coordinates of A. Then (Par(«), <) is isomorphic to the set of PA'S ordered by inclusion. For the Mobius function of (Par(n), <) see Exercise 3.55, and for some further properties see Exercise 7.2. For our final partial order, also on Par(n), it suffices to take any linear order compatible with (i.e., a linear extension of) dominance order. The most convenient R R is reverse lexicographic order, denoted <. Given \X\ = |/x|, define \i < X if either \i = X, or else for some /, /xi = A.i,..., /z; = Xt, /z/+i < Xi+\. For instance, the order > on Par(6) is given by 6 > 51 > 42 > 411 > 33 > 321 > 3111 > 222 > 2211 > 21111 > 111111. On the other hand, in dominance order the partitions 33 and 411, as well as 3111 and 222, are incomparable. We would like to make one additional definition related to partitions. Define the rank of a partition X = (X\, X2,...), denoted rank(A), to be the largest integer i for which Xt > i. Equivalently, rank(A) is the length of the main diagonal of the diagram of X or the side length of the Durfee square of X (defined in the solution to Exercise 1.23(b)). Note that rank(X) = rank(Ar). 7.3 Monomial Symmetric Functions n Given X = (k\, X2,...) h n, define a symmetric function mx(x) e A by mx = a where the sum ranges over all distinct permutations a = (a 1, a2,...) of the entries of the vector k = (k\, A.2,.. .)• For instance, m2 = a n We call mx a monomial symmetric function. Clearly if / = ^2a cax e A then c m / = X^xi-n ^ ^- ^ follows that the set {mx : k \- n} is a (vector space) basis for Downloaded from https://www.cambridge.org/core. Access paid by the UC Berkeley Library, on 26 Jun 2019 at 21:59:31, subject to the Cambridge Core terms of use, availableCambridge at https://www.cambridge.org/core/terms Books Online © Cambridge University. https://doi.org/10.1017/CBO9780511609589.006 Press, 2010 290 7 Symmetric Functions A", and hence that dimA" = p(n), the number of partitions of n. Moreover, the set {mx : X e Par} is a basis for A. 7.4 Elementary Symmetric Functions Next we define the elementary symmetric functions ex for X e Par by the formulas W en = m\n = c/j • • • jCjn, n > 1 (with £o = 0 = 1) (7.2) if X = (X A ,.
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