h r t r 1 (1 x i=1 (1 + x i t). e r t r = i=1 ( 1) i e i h r i = 0 r 1.

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1 . Four definitions of Schur functions.. Lecture : Jacobi s definition (ca 850). Fix λ = (λ λ n and X = {x,...,x n }. () a α = det ( ) x α i j for α = (α,...,α n ) N n (2) Jacobi s def: s λ = a λ+δ /a δ (3) e k = s ( k ) = i <i 2 < <i k x i x i2 x ik (4) h k = s (k) = i i 2 i k x i x i2 x ik.2. Lecture 2: The Jacobi-Trudi Formula. Theorem. [Fundamental Thm Of Symmetric Functions] As rings, Z[x,...,x n ] Sn Z[e,...,e n ]. Proof. Need to show every symmetric polynomial can be written as a polynomial function of the e i s. Any monomial in the e i s is of the form e i e i2 e ij so might as well write indices in decreasing order since they commute. Define e λ := e λ e λ2 e λp for any partition λ = (λ λ p > 0). Consider the expansion of e λ into monomials. Each monomial can be thought of as a filling of λ where row i is filled in increasing order and represents a monomial chosen from e λi for i p. For example, if λ = (3, 2,, ) then two valid fillings are S = T = We use the notation x T to mean the monomial determined by the content of T. In the example, x S = x 4 x 2 2x 3 and x T = x 2 x 2 x 4 x 7 x 2 9. Thus, e λ = e λ e λ2 e λp = fillings T of λ with rows strictly increasing. x T = m λ + µ a λ,µ m µ where the sum is over all partitions µ of size λ which are larger than λ (conjugate of λ) in reverse lexicographic order.

2 2 From this expansion we see that the a λ,µ s are non-negative integers in a matrix with s down the diagonal. This matrix is lower triangular if the partitions of the same size as λ are ordered in rev lex order. Such a matrix is invertible, so the {e λ : λ is a partition with λ n } form a basis of Z[X] Sn. We conclude that every symmetric polynomial is expressible in a unique way in terms of the e i s. This implies Z[X] Sn Z[e,...,e n ] and that the e i s are algebraically independent. Corollary.. The e i s are algebraically independent so they minimally generate the ring of symmetric polynomials. Remark.2. There are relations in Z[x,...,x n ] Sn among the h i s for i since there are an infinite number of them. HW Express h n+ (x,...,x n ) in terms of the h i s for i n. Computer exploration might help here. Interestingly, the h i s become algebraically independent if n approaches infinity. Define the ring Λ = inverse limit as n Z[x,..., x n ] Sn. Thus Z[x,..., x n ] Sn = Λ xi =0 i>n. Define generating functions for the homogeneous and elementary symmetric functions in Λ by H(t) = h r t r = ( x r 0 i t) i= E(t) = r 0 e r t r = ( + x i t). i= Note, H(t) = E( t) =. Equating coefficients of t r we get r (.) ( ) i e i h r i = 0 r. i=0 Define the ring homomorphism ω : Λ Λ by ω(e r ) = h r. By (.), we see that ω is an involution proving the following proposition. Proposition.3. The ring Λ = Z[h, h 2,...] = Z[e, e 2,...]. Going back to the symmetric polynomial situation again, since Schur functions are symmetric they must be expressible as polynomials in the e i s. This gives us another way to define Schur functions.

3 . FOUR DEFINITIONS OF SCHUR FUNCTIONS 3 Theorem 2. [Jacobi-Trudi Formula] (aka Giambelli Formula) Let λ = (λ λ p ) and let q = λ. Then (.2) (.3) s λ = det (h λi i+j) i,j p = det ( ) e λ i i+j i,j q where by definition e i = h i = 0 if i < 0 and e 0 = h 0 =. Example: Assume λ = (4, 2, 2) so λ = (3, 3,, ). Then s λ is the determinant of the matrix e 3 e 4 e 5 e 6 e 2 e 3 e 4 e 5 0 e e e HW 2: Assume A, B SL N (F) are inverses of each other. Then for any two k- subsets I, J [N] we have det(a I,J ) = ( ) Σ(I)+Σ(J) det(b J c I c). Here A I,J is the submatrix of A taking entries only in rows indexed by elements of I and columns indexed by elements of J. Also, write Σ(I) for the sum of the elements in I. Note, the matrix on the left will usually be a different size from the matrix on the right. This identity generalizes the formula for the inverse matrix in terms of cofactors B i,j = ( ) i+j det(a {j} c {i} c). HW 3: For any matrix X = (x i,j ), we have det(x i,j ) = det(( ) i j x i,j ). Proof. Let s prove the second equality first (due to Aitken). Let N be an integer larger than either p or q. N = p + q is a good choice. Set H = (h i j ) 0 i,j N E = ( ( ) i j e i j )0 i,j N. Both E and H are lower triangular with s down the diagonal so det(h) = det(e) =. Furthermore, because of (.) we know E and H are inverses of each other. Thus HW 2 applies. Consider the minor of H indexed by rows I = {λ i +p i : i p} and columns J = {p j : j p}. This minor is exactly (.2). Then by HW 2, we have that this minor is equal to ( ) Σ(I)+Σ(J) times its complementary cofactor in E t. What are the indices for the cofactor in terms of λ? Draw the Ferrers diagram for λ in a p by q rectangle. Starting at the lower left corner, number the up and

4 4 right steps around the SW perimeter of λ by 0,, 2,..., p + q = N. The up steps occur at {λ i + p i : i p}, i. e. λ + δ p. The complementary set are the horizontal steps. These horizontal steps are in bijection with the up steps for λ in the transposed picture. The map is given by complementing the numbers in value so the set is given by I c = {N λ i (q i) : i q} = {p λ i + i : i q}. Similarly, J c = {p + j : j q}. So by HW 2, det (h λi i+j) i,j p = det( ) λ (( ) λ i i+j e λ i i+j ) i,j q Distributing the ( ) λ through each row we get an equivalent form ( ( ) i+j e λ i i+j ) = ( eλ i i+j). by HW 3. Then Now the proof of the first equality: s λ = det (h λi i+j). Let e (k) r = e r (x,..., x k,...,x n ) for each k, r n. Define n E (k) (t) = r=0 e (k) r t r = i k( + x i t). H(t) E (k) ( t) = ( x k t). Picking out the coefficient of t a on both sides gives n h a n+j ( ) n j e (k) n j = xa k. j= Let α = (α,...,α n ) N n (weak composition) and define H α = (h αi n+j) i,j n and M = ( ) ( ) n i e k n i Then H α M = ( ) x α i j. Taking determinants of both sides gives det(h α )det(m) = det ( ) x α i j = aα. We find the det(m) by making a judicious choice of α: det(h δ ) = det (h n i n+j ) = det (h j i ) = i,k n since its upper triangular with s along the diagonal. So det(m) = a δ. Thus a α /a δ = det(h δ ) and taking α = λ + δ gives s λ = a λ+δ a δ = det(h λ+δ ) = det (h λi +n i n+j) = det (h λi i+j) i,k n.

5 . FOUR DEFINITIONS OF SCHUR FUNCTIONS 5 Compare with statement of the theorem. It is a p p matrix. To finish the proof just note the block lower triangular form of the n n matrix here. N.B. The final form of the Jacobi-Trudi determinants does not depend on the number of variables n as long as n is larger than the length of the first row or column of the partition. Thus, the expansion of Schur functions into homogeneous or elementary symmetric functions is stable under the inverse limit. This gives us a way to define Schur functions in Λ. (.4) (.5) Corollary.4. ω(s λ ) = s λ. One more big result comes from the Jacobi s definition of Schur functions. Theorem 3. [Pieri s Formula] s λ e k = s λ h k = µ/λ vertical strip of size k s µ µ/λ horizontal strip of size k where a vertical strip is a skew shape with no two cells in the same row and a horizontal strip has no two cells in a column. For example, if λ = (3, 3, ) then the shapes that occur all with multiplicity one in Pieri s formula are s µ So s (3,3,) e 3 = s (4,4,2) + s (4,4,,) + s (4,3,,,) + s (3,3,2,,) + s (3,3,,,,). N.B. The Pieri formula completely determines the way Schur functions multiply since we know the Jacobi-Trudy formula. In fact, we can use this rule alone to define the Schur functions.

6 6 Proof. Expand ( ) a λ+δ e k = sgn(w)x w(λ+δ) x i x i2 x ik w S n i < <i k n = sgn(w)x w(λ+δ) x w(i )x w(i2 ) x w(ik ) w S n = χ {0,} n χ =k i < <i k n a λ+χ+δ. Note that if any λ + χ + δ appearing the last sum is not strictly decreasing then it must have two equal parts so that term vanishes. The remaining non-vanishing terms all correspond with adding a vertical strip to λ. The second formulation follows by applying ω. HW 4: Say that µ is an even partition if all of its parts are even numbers. Show ( ) ( n ) s µ e k = s λ. µ even k=0 λ

Two Remarks on Skew Tableaux

Two Remarks on Skew Tableaux Richard P. Stanley Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 rstan@math.mit.edu Submitted: 2011; Accepted: 2011; Published: XX Mathematics