AN INVOLUTIVE INTRODUCTION TO SYMMETRIC FUNCTIONS MARK WILDON PREFACE In Autumn 2015 I gave a 10 lecture course on symmetric functions at Royal Holloway, University of London, following a slightly uncon- ventional path that emphasised bijective and involutive proofs. This is an expanded version of the lecture notes. Subsections marked ? are not logically essential. A recurring theme is the combinatorial and al- gebraic meaning of the transition matrices between the various bases of the ring of symmetric functions. For example: Elementary to monomial Gale–Ryser Theorem x1.4 and x5.2 Schur to monomial Kostka Numbers (1.10) and Q24 Schur to power sum Symmetric group characters x5.5 Power sum to monomial Polya’s Cycle Index Theorem x5.7 Outline. In x1 the families of elementary, complete homogeneous and power sum symmetric functions are defined. Schur functions are defined combinatorially, us- ing semistandard tableaux, and shown to be symmetric by the Bender–Knuth in- volution. Motivation comes from combinatorial results including the Gale–Ryser Theorem, MacMahon’s Master Theorem, and the Cycle Index Formula for the symmetric group. The ring of symmetric functions is defined formally and then shown to be an inverse limit of the graded rings of symmetric polynomials. In x2 the Jacobi–Trudi Identity is proved ‘by sufficiently general example’ using a special case of an involution due to Lindstrom¨ and Gessel–Viennot. This is now a standard proof: it may be found in [14, x7.16] or [13, x4.5]. In x3 we switch focus to antisymmetric polynomials, and present the elegant in- volutive proofs in [8] of the Pieri, Young and Murnaghan–Nakayama Rules using Loehr’s abacus model. A textbook account may be found in [9]. These are results on the Schur polynomials, defined as a quotient of two antisymmetric determi- nants, so do not obviously relate to the Schur functions already defined. In x4 we establish the equivalence of the two definitions using the Lascoux–Schutzenberger¨ involution (originally defined in [7]). In x5 we unify the results so far using the Hall inner product and the w involu- tion on the ring of symmetric functions. We then prove the key properties of the characteristic isometry, relating class functions of symmetric groups and symmet- ric polynomials and apply it to prove Polya’s´ Cycle Index Formula. Date: July 2017. The Jacobi–Trudi formula and the Lascoux–Schutzenberger¨ involution extend to Schur functions labelled by skew partitions. This generality adds considerably to their utility, but appeared excessive in the early lectures. It was assumed in the final lecture, where I used the Lascoux–Schutzenberger¨ involution to give a proof of the Littlewood–Richardson Rule. The proof is given as a series of ques- tions in x6, starting with Question 26. Most of these questions are on well-known results or proofs: possible exceptions are Question 7 (generalized derangements), Question 13 (an easy way to go wrong in the proof of the Jacobi–Trudi Identity) and Question 21 (an involutive proof of the Murnaghan–Nakayama Rule). Hints, references or solutions for all the questions are given in the final section. Comments. Comments are very welcome. In particular I gratefully acknowledge extremely detailed comments and corrections from Darij Grinberg on an earlier version of these notes. I also thank Eoghan Mc- Dowell for helpful comments and corrections. Of course I have full responsibility for the remaining errors. 1. INTRODUCTION: DEFINITIONS AND MOTIVATION Lecture 1 1.1. Preliminary definitions. The following definitions are standard. Note in particular that partitions have infinitely many parts, all but finitely many of which are zero. Permutations act on the right. Compositions and partitions. A composition of n 2 N0 is an infinite se- quence (a1, a2,...) such that ai 2 N0 for all i and a1 + a2 + ··· = n. (The term ‘weak composition’ is also used in the literature.) The se- quence elements are called parts. By this definition, there is a unique composition of 0, which we denote ?. If a 6= ?, let `(a) be the maxi- mum r such that ar 6= 0 and let `(?) = 0. If `(a) ≤ N then we write a = (a1,..., aN). A composition a is a partition of n if a1 ≥ a2 ≥ . and a1 + a2 + ··· = n. We write a j= n to indicate that a is a composition of n and l ` n to indicate that l is a partition of n. It is often convenient to use exponents to indicate multiplicities of am a a parts: thus (m , . , 2 2 , 1 1 ) denotes the partition with exactly aj parts equal to j, for each j 2 f1, . , mg. For example, (4, 4, 2, 1, 1, 1, 0, . .) = (4, 4, 2, 1, 1, 1) = (42, 30, 21, 13). The Young diagram [l] of a partition l is the set f(i, j) : i, j 2 N, 1 ≤ i ≤ `(l), 1 ≤ j ≤ lig. We represent Young diagrams by diagrams, such as the one shown below for (4, 2, 1, 1): . The conjugate of a partition l is the partition l0 defined by 0 lj = jfi : li ≥ jgj. 3 0 By definition lj = r if and only if l has exactly r parts of size j or more, or equivalently, if and only if column j of [l] has length r. Thus [l0] is obtained from [l] by reflection in the main diagonal. In particular l00 = l for any partition l. Orders on partitions. The dominance order on partitions of n, denoted ¤, is defined by l ¤ m if and only if l1 + ··· + lc ≥ m1 + ··· + mc for all c 2 N. It is a partial order: for example (3, 1, 1, 1) and (2, 2, 2) are incomparable. It is usually the right order to use when working with symmetric functions or symmetric groups. The lexicographic order, denoted >, is defined by l > m if and only if l1 = m1,..., lc−1 = mc−1 and lc > mc for some c. It is a total order refining the dominance order. (The word ‘refining’ is mathematically correct, but may give the wrong impression: more precise information comes from using the dominance order.) Symmetric group. Let Sym(X) denote the symmetric group on a set X. Let Symn = Sym(f1, . , ng). In these notes permutations act on the right. For example, the composition of the cycles (12) and (123) in the symmetric group Sym3 is (12)(123) = (13), and the image of 1 under the permutation (12) is 1(12) = 2. 1.2. Gale–Ryser Theorem. We begin with a combinatorial result. An b a × b matrix X with entries in C has row sums (∑j=1 Xij)i2f1,...,ag and a column sums (∑i=1 Xij)j2f1,...,bg. A matrix is 0-1 if all its entries are either 0 or 1. Theorem 1.1 (Gale–Ryser). Let l and m be partitions of n. There is a 0-1 matrix X with row sums l and column sums m if and only if l0 ¤ m. For example, take l = (4, 1, 1), so l0 = (3, 1, 1, 1). If m = (2, 2, 1, 1) then a suitable 0-1 matrix is 01 1 1 11 @1 0 0 0A 0 1 0 0 while if m = (2, 2, 2) then no such matrix exists. By Question 2, we have l0 ¤ m if and only if l ¢ m0. The condition in the Gale–Ryser Theorem is therefore symmetric with respect to l and m, as expected. Proof that the Gale–Ryser condition is necessary. Let a = `(l) and let b = `(m). Suppose that X is an a × b matrix with row sums l and column sums m. Think of (l1,..., la) as the sizes of a vehicles, and (m1,..., mb) as the sizes of b families. Imagine putting someone from family j into 4 vehicle i if and only if Xij = 1. We then have a way to dispatch the fami- lies so that no members of the same family share a vehicle. Consider the m1 + ··· + mk people in the first k families. They occupy at most k seats in each vehicle, so, looking at the first k columns of the Young diagram 0 0 of [l], we see that l1 + ··· + lk ≥ m1 + ··· + mk, as required. We will shortly use this result to relate elementary symmetric func- tions and monomial symmetric functions. We later use symmetric func- tions to prove that the Gale–Ryser condition is sufficient: see Lemma 5.4. Constructive proofs also exist: see Question 11. 1.3. The ring of symmetric functions L. Given a composition a of n, a a1 a2 define the monomial x = x1 x2 . .. This should be regarded purely formally for the moment. For each n 2 N0 define Cb[x1, x2,...]n to be the C-vector space of all formal infinite C-linear combinations of the xa, for a a composition of n. In symbols a Cb[x1, x2,...]n = f ∑ cax : ca 2 Cg. aj=n ? Note that x = 1, so Cb[x1, x2,...]0 = C. Let ¥ M Cb[x1, x2,...] = Cb[x1, x2,...]n. n=0 Then Cb[x1, x2,...] is a ring, graded by degree, with product defined by formal bilinear extension of xaxb = xa+b, where a + b is the composi- tion defined by (a + b)i = ai + bi for each i 2 N. (The hat is included to distinguish Cb[x1, x2,...] from the polynomial ring in the variables x1, x2 . , which it properly contains. See x1.7 below.) The symmetric group Sym(N) acts as a group of linear transforma- tions of Cb[x1, x2,...] by linear extension of xis = xis. We define Sym(N) L = Cb[x1, x2,...] to be the set of fixed points.