Symmetric functions Contents 1 Preliminaries 1 2 Monomial symmetric functions 2 2.1 Multiplying Monomials . 2 2.2 Dominance order . 3 3 Elementary symmetric functions 3 4 Homogeneous symmetric functions: hλ 3 4.1 Generating Functions . 4 5 Power symmetric functions: pλ 4 5.1 Generating Function: . 4 6 Schur functions 5 6.1 Skew Schur functions . 5 6.2 Symmetric group action . 5 6.3 Schur functions . 6 6.4 The Pieri Rule . 6 7 Littlewood-Richardson coefficients 6 7.1 The bi-alternant formula . 7 7.2 Littlewood-Richardson coefficients . 8 7.3 RSK-Correspondence and the Cauchy identity . 8 8 Jacobi-Trudi Determinants 9 1 Preliminaries Let R be a commutative ring and x1, . , xn a set of indeterminants. Let α = (α1, . , αn) be a vector of non-negative integer coefficients. The set of all α form a monoid under addition which is isomorphic to the monoid of all monomials xα under multiplication: α β α+β α α αn x x = x with x = x1 ··· xn . The corresponding algebra is the ring of polynomials, denoted R[x1, . , xn], consisting of all polyno- mials in n variables with coefficients in R. α We write |α| = deg x = α1 + ··· + αn. An element of R[x1, . , xn] of the form X α f(x1, . , xn) = cαx |α|=d is called a homogeneous polynomial of degree d. With set of all such polynomials denoted by d R[x1, . , xn] , R[x1, . , xn] is a graded ring: M d R[x1, . , xn] = R[x1, . , xn] d≥0 1 There is a natural degree-preserving Sn action on polynomials, where elements of the symmetric group act by permuting the variables. That is, given P (x) ∈ R[x1, . , xn] and σ ∈ Sn: σP (x1, x2, . , xn) = P (xσ1 , xσ2 , . , xσn ) . 2 2 For example, σ = (1 3 2) =⇒ σ x1 + 2x1x3 + x2 = x1 + 2x1x2 + x3 . A polynomial is called symmetric if it is invariant under the action of Sn. The set of symmetric d polynomials of degree d in n-variables is denoted Λn and we set, M d Λn = Λn . d Λn is a subring of R[x1, . , xn], called the ring of symmetric polynomials. Examples include: x1 + 2x1x2x3 + x2 + x3 ∈ Λ3 3 x1x2x3 + x1x2x4 + x2x3x4 + x1x3x4 ∈ Λ4 2 Monomial symmetric functions The most obvious symmetric polynomials are constructed by symmetrizing the monomial term xλ, for a partition λ = (λ1, . , λn) of weakly decreasing non-negative integers. The monomial symmetric functions: mλ X β β β1 β2 βn mλ = x where x = x1 x2 ··· xn β:β∗=λ over all distinct β where β∗ is the partition rearrangement of β. For example, with λ = (2, 1, 1) and n = 4: (2, 2, 1, 0) (2, 2, 0, 1) (1, 2, 2, 0) (1, 2, 0, 2) (2, 1, 2, 0) (2, 1, 0, 2) (0, 2, 2, 1) (0, 2, 1, 2) (0, 1, 2, 2) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 m2,1,1 = x1x2x3 + x1x2x4 + x1x2x3 + x1x2x4 + x1x2x3 + x1x2x4 + x2x3x4 + x2x3x4 + x2x3x4 Furthermore, the mλ are clearly independent and if the homogeneous polynomial X α f = cαx |α|=d is symmetric, then the cα must remain constant as α ranges over the Sn orbit. Thus, the set of all mλ d with |λ| = d spans Λn. Note if `(λ) > n, we don’t have enough variables and mλ = 0. To avoid this issue, we instead work in the vector space spanned by all the mλ called the ring of symmetric functions: Λ = R[mλ] . d Λ is closed under product, and is a graded ring. If Λ is the space spanned by all mλ of degree d, M Λ = Λd . d≥0 d Theorem 1. {mλ : |λ| = d} is a basis of Λ Corollary 2. The dimension of Λd is the number of partitions of d. 2 2.1 Multiplying Monomials The study of the product of certain monomial symmetric functions will later help us understand other bases for Λ. More precisely, we are interested in determining the possible terms in the right hand side of X ν k m1k mλ = cλ1k mν where 1 = (1,..., 1) . |ν|=|λ|+k Note that since the product of symmetric functions is symmetric, ν ν cλ1k = the coefficient of mν in mλm1k = the coefficient of x in mλm1k α ν Since the monomial terms in mλ are x where α rearranges to λ, cλ1k is the number of ways to fill rows of ν with at most one row of λ and of (1,..., 1). For example, filling shapes with (2,1) and (1,1) gives −→ −→ −→ −→ m1,1m2,1 = 2m2,2,1 + m3,2 + 2m3,1,1 + 3m2,1,1,1 It turns out that the only shapes with at least one filling of λ and 1k are those obtained by adding a vertical k-strip to the diagram of λ. + a vertical 2-strip = , , , To convince yourself of this fact, note first that since ν is obtained from the parts of λ and 1k, each row of ν must be λi or λi + 1 for some i. Thus, the length of the longest row will be at most λ1 + 1 and at least λ1 implying ν1 contains λ1. Similarly, the length of the second longest row will be λ2 + 1 or λ2 implying ν2 contains λ2, etc. 2.2 Dominance order We put a partial order C on the set of all partitions (and consequently on the set of monomial symmetric functions mµ C mλ when µ C λ) called dominance order by declaring that λ B µ ⇐⇒ λ1 + ··· + λj ≥ µ1 + ··· + µj for all j Many bases are triangularly related with respect to B. Intuitively, λ is greater than µ in dominance order if the Ferrers diagram of λ is wider than µ. 0 0 Remark 3. λ C µ ⇐⇒ λ B µ We can use dominance order to refine our result on the product of monomials: X ν X ν m1k mλ = cλ,1k mν = mλ+1k + cλ,1k mν (1) k ν=λ+vertical k-strip ν C λ+1 k where λ + 1 = (λ1 + 1, . , λk + 1, λk+1,...). 3 Elementary symmetric functions The elementary symmetric functions are eλ = eλ1 eλ2 ··· eλ` where X ek(x1, . , xn) = xi1 xi2 ··· xik and e0 = 1 . 1≤i1<i2<···<ik≤n For example, e1 = x1 + x2 + ··· and e2 = x1x2 + x1x3 + ··· . Clearly m1k = ek. Note the degree(ek) = k and degree(eλ = eλ1 ··· eλ` ) = |λ|. Further, when k > n, ek = 0. Theorem 4. {eµ} forms a basis for Λ. 3 Proof. Homework. a1 a2 ∞ Since the set {eλ} consists of all monomials e1 e2 ··· where ai ∈ N , and any element of Λ can be expressed as a linear combination of eλ, every element of Λ is uniquely expressible as a polynomial in the er and the er are algebraically independent. ∼ Theorem 5. Λ = Q[e1, e2,...] 4 Homogeneous symmetric functions: hλ Definition: Let h0 = 1 and hλ = hλ1 hλ2 ··· hλ` where X X hk(x1, . , xn) = mλ = xi1 xi2 ··· xik |λ|=k 1≤i1≤i2≤···≤ik≤n Note that h1 = e1. Sometimes hk is called the complete symmetric function since it is the sum over all P P 2 P 2 2 monomials: h1 = xi and h2 = xi + xixj = x1 + x2 + x1x2 + ··· . The homogeneous functions are not triagulary related to the monomials. We shall thus appeal to the use of generating functions to show that the homogeneous symmetric functions provide a basis for Λ. 4.1 Generating Functions Define X k X k E(t) = ek t H(t) = hk t k≥0 k≥0 Theorem 6. Y Y 1 E(t) = (1 + xit) H(t) = 1 − xit i≥1 i≥1 Proof. Y X X 2 (1 + xit) = (1 + tx1)(1 + tx2) ··· = 1 + txi + t xixj + ··· i≥1 i≥1 i<j The identity for H(t) follows similarly, starting with the geometric series expansion. d Theorem 7. {hλ : λ ` d} is a basis for Λ . Proof. Since the number elements hλ is the number of partitions of d, it suffices to show that they generate the eµ. Further, since both hλ and eλ are multiplicative we shall simply show that ek = f(h1, . , hk) for 1 some polynomial f. Our identies reveal that H(t) = E(−t) , or H(t)E(−t) = 1. Substituting hk and e` in the summations gives ! ! X k X l t hk (−t) el = 1 , k l and by comparing coefficients of tn in both sides we see that X ` (−1) hke` = 0 n 6= 0 (2) k+l=n Thus, en = h1en−1 − h2en−2 + · · · ± hn−1e1 which is a polynomial in the h’s by induction on n. 4 5 Power symmetric functions: pλ X k pk = xi and pλ = pλ1 ··· pλ` i r r p1 = x1 + x2 + ··· = e1 = h1 pr = x1 + x2 + ··· = m(r) d Theorem 8. {pλ : λ ` d} is a basis of Λ P Proof. Let pλ = µ cλµmµ. If we can show that cλµ = 0 for all µ ≥ λ and cλλ 6= 0 then the transition µ1 µ2 µn λ1 λ1 λ2 matrix is invertible and then pλ must be a basis. If x1 x2 ··· xn appears in pλ = (x1 +x2 +··· )(x1 + λ2 x2 + ··· ) ··· , then each µi is a sum of λj’s. Since adding together parts of a partition makes it become larger in dominance order, mλ must be the smallest term that occurs. 5.1 Generating Function: Studying the generating function for the power sums reveals: Proposition 9. def X tn P (t) p = ln H(t) = n n n≥1 1 Proof.