The Kronecker Product of Schur Functions Indexed by Two-Row Shapes or Hook Shapes

Transcription

1 Journal of Algebraic Combinatorics 4 (00), c 00 Kluwer Academic Publishers. Manufactured in The Netherlands. The Kronecker Product of Schur Functions Indexed by Two-Row Shapes or Hook Shapes MERCEDES H. ROSAS mrosas@usb.ve Departamento de Matemáticas, Universidad Simón Bolívar, Apdo, Postal 89000, Caracas, Venezuela Received July 9, 999; Revised October 6, 000 Abstract. The Kronecker product of two Schur functions s µ and s ν, denoted by s µ s ν, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions µ and ν. The coefficient of s λ in this product is denoted by γµν λ, and corresponds to the multiplicity of the irreducible character χ λ in χ µ χ ν. We use Sergeev s Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for s λ [XY] to find closed formulas for the Kronecker coefficients γµν λ when λ is an arbitrary shape and µ and ν are hook shapes or two-row shapes. Remmel (J.B. Remmel, J. Algebra 0 (989), 00 8; Discrete Math. 99 (99), 65 87) and Remmel and Whitehead (J.B. Remmel and T. Whitehead, Bull. Belg. Math. Soc. Simon Stiven (994), ) derived some closed formulas for the Kronecker product of Schur functions indexed by two-row shapes or hook shapes using a different approach. We believe that the approach of this paper is more natural. The formulas obtained are simpler and reflect the symmetry of the Kronecker product. Keywords: Kronecker product internal product, Sergeev s formula. Introduction The aim of this paper is to derive an explicit formula for the Kronecker coefficients corresponding to partitions of certain shapes. The Kronecker coefficients, γµν λ, arise when expressing a Kronecker product (also called inner or internal product), s µ s ν, of Schur functions in the Schur basis, s µ s ν = λ γ λ µν s λ. () These coefficients can also be defined as the multiplicities of the irreducible representations in the tensor product of two irreducible representations of the symmetric group. A third way to define them is by the comultiplication expansion. Given two alphabets X = x + x + and Y = y + y +, expressed as the sum of its elements, the comultiplication expansion is given by s λ [XY] = γµν λ s µ[x]s ν [Y ], () µ,ν

2 54 ROSAS where s λ [X] means s λ (x, x,...) and s λ [XY] means s λ (x y, x y,...,x i y j,...). Remmel [9, 0] and Remmel and Whitehead [] have studied the Kronecker product of Schur functions corresponding to two two-row shapes, two hook shapes, and a hook shape and a two-row shape. We will use the comultiplication expansion () for the Kronecker coefficients, and a formula for expanding a Schur function of a difference of two alphabets due to Sergeev [, 4] to obtain similar results in a simpler way. We believe that the formulas obtained using this approach are elegant and reflect the symmetry of the Kronecker product. In the three cases we found a way to express the Kronecker coefficients in terms of regions and paths in N.. Basic definitions A partition λ of a positive integer n, written as λ n, is an unordered sequence of natural numbers adding to n. We write λ as λ = (λ,λ,...,λ n ), where λ λ...,and consider two such strings equal if they differ by a string of trailing zeroes. The nonzero numbers λ i are called the parts of λ, and the number of parts is called the length of λ, denoted by l(λ). In some cases, it is convenient to write λ = ( d d n d n ) for the partition of n that has d i equals to i. Using this notation, we define the integer z λ to be d d! d d! n d n d n!. We identify λ with the set of points (i, j) in N defined by j λ i, and refer to them as the Young diagram of λ. The Young diagram of a partition λ is thought of as a collection of boxes arranged using matrix coordinates. For instance, the Young diagram corrresponding to λ = (4, 3, ) is To any partition λ we associate the partition λ, its conjugate partition, defined by λ i = {j : λ j i}. Geometrically, λ can be obtained from λ by flipping the Young diagram of λ around its main diagonal. For instance, the conjugate partition of λ = (4, 3, ) is λ = (3,,, ), and the corresponding Young diagram is We recall some facts about the theory of representations of the symmetric group, and about symmetric functions. See [7] or [3] for proofs and details. Let R(S n ) be the space of class function in S n, the symmetric group on n letters, and let n be the space of homogeneous symmetric functions of degree n. A basis for R(S n ) is given by the characters of the irreducible representations of S n. Let χ µ be the irreducible character of S n corresponding to the partition µ. There is a scalar product, Sn on R(S n )

3 KRONECKER PRODUCT OF SCHUR FUNCTIONS 55 defined by χ µ,χ ν Sn = χ µ (σ )χ ν (σ ), n! σ S n and extended by linearity. A basis for the space of symmetric functions is given by the Schur functions. There exists a scalar product, n on n defined by s λ, s µ n = δ λµ, where δ λµ is the Kronecker delta, and extended by linearity. Let p µ be the power sum symmetric function corresponding to µ, where µ is a partition of n. There is an isometry ch n : R(S n ) n, given by the characteristic map, ch n (χ) = µ n z µ χ(µ)p µ. This map has the remarkable property that if χ λ is the irreducible character of S n indexed by λ, then ch n (χ λ ) = s λ, the Schur function corresponding to λ. In particular, we obtain that s λ = µ n z µ χ λ (µ)p µ. Finally, we use the fact that the power sum symmetric functions form an orthogonal basis satisfying that p λ, p µ n = z µ δ λµ to obtain χ λ (µ) = s λ, p µ. (3) Let λ, µ, and ν be partitions of n. The Kronecker coefficients γµν λ are defined by γµν λ = χ λ,χ µ χ ν Sn = χ λ (σ )χ µ (σ )χ ν (σ ). (4) n! σ S n Equation (4) shows that the Kronecker coefficients γµν λ are symmetric in λ, µ, and ν. The relevance of the Kronecker coefficients comes from the following fact: Let X µ be the representation of the symmetric group corresponding to the character χ µ. Then χ µ χ ν is the character of X µ X ν, the representation obtained by taking the tensor product of X µ and X ν. Moreover, γµν λ is the multiplicity of X λ in X µ X ν. Let f and g be homogeneous symmetric functions of degree n. The Kronecker product, f g,isdefined by f g = ch n (uv), (5) where u = (ch n ) ( f ), and v = (ch n ) (g), and uv(σ) = u(σ )v(σ ). To obtain () from this definition, we set f = s µ, g = s ν, u = χ µ, and v = χ ν in (5).

4 56 ROSAS The Kronecker product has the following symmetries: s µ s ν = s ν s µ. s µ s ν = s µ s ν. Moreover, if λ is a one-row shape γ λ µν = δ µ,ν. We introduce the operation of substitution or plethysm into a symmetric function. Let f be a symmetric function, and let X = x + x + be an alphabet expressed as the sum of its elements. We define f [X] by f [X] = f (x, x,...). In general, if u is any element of Q[[x, x,...]], we write u as α c αu α where u α is a monomial with coefficient. Then p λ [u] isdefined by setting p n [u] = α c α u n α p λ [u] = p λ [u] p λn [u] for λ = (λ,...,λ n ).Wedefine f [u] for all symmetric functions f by saying that f [u] is linear in f. Let X = x + x + and Y = y + y + be two alphabets as the sum of their elements. We define their sum by X + Y = x + x + +y + y +, and the product by XY = x y + +x i y j +. Then p n [X + Y ] = p n [X] + p n [Y ], p n [XY] = p n [X]p n [Y ]. (6) The inner product of function in the space of symmetric functions in two infinite alphabets is defined by, XY =, X, Y, where for any given alphabet Z,, Z denotes the inner product of the space of symmetric functions in Z. For all partitions ρ, we have that p ρ [XY] = p ρ [X]p ρ [Y ]. If we rewrite (3) as p ρ = λ χ λ (ρ)s λ, then χ λ s λ [XY] = χ µ χ ν s µ [X]s ν [Y ]. (7) µ,ν λ

5 KRONECKER PRODUCT OF SCHUR FUNCTIONS 57 Taking the coefficient of χ λ on both sides of the previous equation we obtain s λ [XY] = χ λ,χ µ χ ν s µ [X]s ν [Y ]. Finally, using the definition of Kronecker coefficients (4) we obtain the comultiplication expansion (). Notation Let p be a point in N. We say that (i, j) can be reached from p, written p (i, j), if(i, j) can be reached from p by moving any number of steps south west or north west, when we use the coordinate axes as it is usually done in the cartesian plane. We define the weight function ω by { x i y j, if p (i, j), ω p (i, j) = 0, otherwise. Notation We denote by x the largest integer less than or equal to x and by x the smallest integer greater than or equal to x. If f is a formal power series, then [x α ] f denotes the coefficient of x α in f. Following Donald Knuth we denote the characteristic function applied to a proposition P by enclosing P with brackets, ((P)) = {, if proposition P is true, 0, otherwise. We use double brackets to distinguish between the Knuth s brackets and the standard ones. 3. The case of two two-row shapes The object of this section is to find a closed formula for the Kronecker coefficients when µ = (µ,µ ) and ν = (ν,ν ) are two-row shapes, and when we do not have any restriction on the partition λ. We describe the Kronecker coefficients γ λ µν in terms of paths in N. More precisely, we define two rectangular regions in N using the parts of λ. Then we count the number of points in N inside each of these rectangles that can be reached from (ν,µ +), if we are allowed to move any number of steps south west or north west. Finally, we subtract these two numbers. We begin by introducing two lemmas that allow us to state Theorem in a concise form. Note that we use the coordinate axes as it is usually done in the cartesian plane. Lemma Let k and l be positive numbers. Let R be the rectangle with width k, height l, and lower left square (0, 0).Define σ k,l (h) = {(u,v) R N : (h, 0) (u,v)}

6 58 ROSAS Figure. The definition of σ. Then 0, if h < 0 ( ) h +, if 0 h < min(k, l) ( ) h s σ k,l (s) + min(k, l), if min(k, l) h < max(k, l) σ k,l (h) = kl σ k,l (k + l h 4), if h is even and max(k, l) h kl σ k,l (k + l h 4), if h is odd and max(k, l) h where s is defined as follows: If h min(k, l) is even, then s = min(k, l) ; otherwise s = min(k, l). Example By definition σ 9,5 (4) counts the points in N in figure marked with. Then σ 9,5 (4) = 9. Similarly, σ 9,5 (8) counts the points in N in figure marked either with the symbol or with the symbol. Then σ 9,5 (8) = 9. Proof: If h is to the left of the 0th column, then we cannot reach any of the points in N inside R. Hence, σ k,l (h) should be equal to zero. If 0 h min(k, l), then we are counting the number of points of N that can be reached from (h, 0) inside the square S of side min(k, l). We have to consider two cases. If h is odd, then we are summing (h + ) = ( h + ). On the other hand, if h is even, then we are summing (h + ) = ( h + ). If min(k, l) h < max(k, l), then we subdivide our problem into two parts. First, we count the number of points of N that can be reached from (h, 0) inside the square S by σ k,l (s). Then we count those points of N that are in R but not in S. Since h < max(k, l) all diagonals have length min(k, l) and there are h s of them. See figure for an example. If max(k, l) h, then it is easier to count the total number of points of N that can be reached from (h, 0) inside R by choosing another parameter ĥ big enough and with the same parity as h. Then we subtract those points of N in R that are not reachable from (h, 0) because h is too close. So, if ĥ is even this number is kl/.ifĥ is odd this number is kl/. Then we subtract those points that we should not have counted. We express this number in terms of the

7 KRONECKER PRODUCT OF SCHUR FUNCTIONS 59 function σ. The line y = x + h + intersects the line y = l atx = h l + 3. This is the x coordinate of the first point on the last row that is not reachable from (h, 0). Then to obtain the number of points that can be reached from this point by moving south west or north west, but that were not supposed to be counted, we subtract h l + 3tok. We have obtained that are σ k,l (k + l h 4) points that we should not have counted. Note that σ k,l is symmetrical on k and l Lemma Let a, b, c, and d be nonnegative integers. Let R be the rectangle with vertices (a, c), (a + b, c), (a, c + d), and (a + b, c + d). Wedefine Then Ɣ(a, b, c, d)(x, y) = {(u,v) R N : (x, y) (u,v)}. Ɣ(a, b, c, d)(x, y) σ b +,d + (x + y a c), = σ b +,y c + (x a) + σ b +,c + d y + (x a) δ, σ b +,d + (x y + c + d a), 0 y c c < y < c + d c + d y where δ is defined as follows If x < a, then δ = 0. Ifa x a + b, then δ = x a+. Finally, if x > a + b then we consider two cases: If x a b is even then δ = b+ ; otherwise, δ = b+. Proof: We consider three cases. Note that the letter c indicates the height of the base of the rectangle. If 0 y c then the first position inside R that we reach is (x + y a c, c). Therefore, we assume that we are starting at this point. Similarly, if y c + d, then the first position inside R that we reach is (x y + c + d a, c). Again, we can assume that we are starting at this point. On the other hand, if c < y < c + d, we are at a point whose height meets the rectangle. We subdevide the problem in two parts. The number of positions to the north of us is counted by σ b+,y c+ (x a). The number of positions to the south of us is counted by σ b+,c+d y+ (x a). Wedefine δ to be the number of points of N that we counted twice during this process. Then it is easy to see that δ is given by the previous definition. To compute the coefficient u ν in the expansion f [X] = η u ηs η [X] for f, itis enough to expand f [x + +x n ] = η u ηs η [x + +x n ] for any n l(ν). (See [7, Section I.3], for proofs and details.) Therefore, in this section we work with symmetric functions in a finite number of variables. Jacobi s definition of a Schur function on a finite alphabet X = x + x + +x n as a quotient of alternants says that s λ [X] = s λ (x,...,x n ) = det( x λ j +n j) i i, j n i< j (x. (8) i x j )

8 60 ROSAS By the symmetry properties of the Kronecker product it is enough to compute the Kronecker coefficients γ λ µν when ν µ. Theorem Let µ, ν, and λ be partitions of n, where µ = (µ,µ ) and ν = (ν,ν ) are two two-row partitions and let λ = (λ,λ,λ 3,λ 4 ) be a partition of length less than or equal to 4. Assume that ν µ. Then γ λ µν = (Ɣ(a, b, a + b +, c) Ɣ(a, b, a + b + c + d +, c))(ν,µ + ). where a = λ 3 + λ 4, b = λ λ 3, c = min(λ λ,λ 3 λ 4 ) and d = λ + λ 4 λ λ 3. Proof: Set X = + x and Y = + y in the comultiplication expansion () to obtain s λ [( + y)( + x)] = γ λ µν s µ[ + y]s ν [ + x]. (9) Note that the Kronecker coefficients are zero when l(λ) > 4. The idea of the proof is to use Jacobi s definition of a Schur function as a quotient of alternants to expand both sides of the previous equation, and then get the Kronecker coefficients by looking at the resulting expansions. Let ϕ be the polynomial defined by ϕ = ( x)( y)s λ [( + y)( + x)] = ( x)( y)s λ (, y, x, xy). Using Jacobi sdefinition of a Schur function we obtain y λ +3 y λ + y λ 3+ y λ 4 x λ +3 x λ + x λ 3+ x λ 4 (xy) λ+3 (xy) λ+ (xy) λ3+ (xy) λ 4 ϕ =. (0) xy( xy)(y x)( x)( y) On the other hand, we may use Jacobi s definition to expand s µ [ + y] and s ν [ + x]. Substitute this results into (9): s λ [( + y)( + x)] = µ=(µ,µ ) ν=(ν,ν ) = µ=(µ,µ ) ν=(ν,ν ) γ λ µν γ λ µν ( y µ y µ )( + x ν x ν ) + y x x ν y µ x ν y µ + x ν+ y µ + x ν+ y µ +. ( x)( y) Since ν + and µ + are both greater than n, Eq. () implies that the coefficient of x ν y µ in ϕ is γµν λ. It is convenient to define an auxiliary polynomial by ζ = ( xy)(y x)ϕ. () ()

9 KRONECKER PRODUCT OF SCHUR FUNCTIONS 6 Let ξ be the polynomial defined by expanding the determinant appearing in (0). Equations (0) and () imply ζ = ξ xy( x)( y). Let ξ i, j be the coefficient of x i y j in ξ. (Note that ξ i, j is zero if i < or j <, because ξ is a polynomial divisible by xy.) Let ζ i, j be the coefficient of x i y j in ζ. Then ζ i, j x i y j = ξ i, j x i y j = xy( x)( y) i, j 0 i, j 0 i, j,k,l 0 ξ i k, j l x i y j. (3) Comparing the coefficient of x i y j on both sides of Eq. (3) we obtain that ζ i, j = i j ξ i+ k, j+ l = ξ k+,l+ (4) k,l 0 k=0 l=0 We compute ζ i, j from (4) by expanding the determinant appearing on (0). We consider two cases. Case. Suppose that λ + λ 4 >λ + λ 3. Then λ + λ + 4 >λ + λ >λ + λ 4 + λ + λ 3 + >λ + λ 4 + >λ 3 + λ 4. We record the values of ξ j+,i+ in Table. We use the convention that ξ i+, j+ is zero whenever the (i, j) entry is not in Table. Equation (4) shows that the value of ζ i, j can be obtained by adding the entries north west of the point (i, j) in Table. In Table we record the values of ζ i, j. Table. The values of ξ j+,i+ when λ + λ 4 λ + λ 3. i\ j λ 3 + λ 4 λ + λ 4 + λ + λ 3 + λ + λ 4 + λ + λ λ + λ + 4 λ 3 + λ λ + λ λ + λ λ + λ λ + λ λ + λ

10 6 ROSAS Table. The values of ζ i, j when λ + λ 4 λ + λ 3. i\ j I I I 3 I 4 I 5 I 6 I 7 I I I I I I I Where Case. I = [0,λ 3 + λ 4 ), I = [λ 3 + λ 4,λ + λ 4 ], I 3 = [λ + λ 4 +,λ + λ 3 + ], I 4 = [λ + λ 3 +,λ + λ 4 + ], I 5 = [λ + λ 4 +,λ + λ 3 + ], I 6 = [λ + λ 3 + 3,λ + λ + 3], I 7 = [λ + λ + 4, ]. Suppose that λ + λ 4 λ + λ 3. Then λ + λ + 4 >λ + λ >λ + λ 3 + >λ + λ 4 + >λ + λ 4 + >λ 3 + λ 4. Note that in Table, the rows and columns corresponding to λ + λ 4 + and λ + λ 3 + are the same. Therefore, the values of ξ i, j for λ + λ 4 λ + λ 3 are recorded in Table, if we set I 3 = [λ + λ 4 +,λ + λ 4 + ] I 4 = [λ + λ 4 +,λ + λ 3 + ] I 5 = [λ + λ 3 +,λ + λ 3 + ], and define the other intervals as before. In both cases, let ϕ i, j be the coefficient of x i y j in ϕ. Using () we obtain that ϕ = ζ i, j x i y j ( xy)(y x) i, j 0 = ζ i l, j l x i y j y x i, j,l 0 = ζ i k l, j+k l+ x i y j. (5) i, j,k,l 0

11 KRONECKER PRODUCT OF SCHUR FUNCTIONS 63 Figure. The right-most point in figure has coordinates (, ). (Note: We can divide by y x because ϕ = 0 when x = y.) Comparing the coefficients of x i y j on both sides of Eq. (5), we obtain ϕ i, j = k,l 0 ζ i k l, j+k l+. Therefore, ϕ ν,µ = ν i, j=0 ζ ν i j,µ +i j+. (6) We have concluded that the coefficient of x ν y µ in ϕ is γ λ µ,ν. Hence γ λ µν = ϕ ν,µ can be obtained by adding the entries in Table at all points of N that can be reached from (ν,µ + ). after we flip Table around the horizontal axis. See figure. By hypothesis ν µ n/. Then, if we start at (ν,µ +) and move as previously described, the only points of N that we can possibly reach and that are nonzero in Table are those in I I 3 or I I 5. Hence, we have that ϕ ν,µ is the number of points of N inside I I 3 that can be reached from (ν,µ + ) minus the ones that can be reached in I I 5. All other entries that are reachable from (ν,µ + ) are equal to zero. Case. The inequality λ + λ 4 >λ + λ 3 implies that λ + λ 4 + > t n. Moreover, µ ν implies that we are only considering the region of N given by 0 i j n. The number of points of N that can be reached from (ν,µ + ) inside I I 3 is given by Ɣ(λ 3 + λ 4,λ λ 3,λ + λ 4 +,λ 3 λ 4 ). Similarly, the number of points of N that can be reached from (ν,µ + ) inside I I 5 is given by Ɣ(λ 3 + λ 4,λ λ 3,λ + λ 4 +, λ 3 λ 4 ). Case. The inequality λ + λ 3 λ + λ 4, implies that λ + λ 4 + > n. Moreover, µ ν implies that we are only considering the region of N given by 0 i j n. The number of points of N that can be reached from (ν,µ + ) inside I I 3 is given by Ɣ(λ 3 + λ 4,λ λ 3,λ + λ 4 +,λ λ ). Similarly, the number of points of N that can be reached from (ν,µ + ) inside I I 5 is given by Ɣ(λ 3 + λ 4,λ λ 3,λ + λ 3 +,λ λ ). Corollary Let µ = (µ,µ ), ν = (ν,ν ), and λ = (λ,λ ) be partitions of n. Assume that ν µ λ. Then γ λ µν = (y x)(y x), where x = max(0, µ +ν +λ n ) and y = µ +ν λ +.

12 64 ROSAS Proof: Set λ 3 = λ 4 = 0 in Theorem. Then we notice that the second possibility in the definition of Ɣ, that is, when c < y < c + d, never occurs. Note that ν + µ λ ν + µ λ for all partitions µ, ν, and λ. Therefore, γ λ µν = σ λ +,(ν + µ λ ) σ λ +,(ν + µ λ ). Suppose that ν + µ λ < 0. By Lemma, σ λ +,(h) = 0 when h < 0. Hence, we obtain that γµν λ = 0. Therefore, in order to have γ µν λ not equal to zero, we should assume that ν + µ λ 0. If 0 ν + µ λ <λ +, then ν + µ λ + σ λ +,(ν + µ λ ) = Similarly, if 0 ν + µ λ <λ +, then ν + µ + λ n σ λ +,(ν + µ λ ) = It is easy to see that all other cases obtained in Lemma for the computation of σ k,l can not occur. Therefore, defining x and y as above, we obtain the desired result. Example If µ = ν = λ = (l, l) or µ = ν = (l, l) and λ = (3l, l), then from the previous corollary, we obtain that l + l γµν λ = = ((l is even)) Note that to apply Corollary to the second family of shapes, we should first use the symmetries of the Kronecker product. Corollary unbounded. The Kronecker coefficients γµν λ, where µ and ν are two-row partitions, are Proof: It is enough to construct an unbounded family of Kronecker coefficients. Assume that µ = ν = λ = (3l, l). Then from the previous corollary we obtain that l + γµν λ = 4. Sergeev s formula The fundamental tool for the study of the Kronecker product on the remaining two cases is Sergeev s formula for the difference of two alphabets. See [, 4], or [7, section I.3]

13 KRONECKER PRODUCT OF SCHUR FUNCTIONS 65 for proofs and comments. In order to state Sergeev s formula we need to introduce some definitions. Definition Let X m = x + +x m be a finite alphabet, and let δ m = (m, m,...,, 0).Wedefine X δ m m by X δ m m = x m x m. Definition An inversion of a permutation α α α n is a pair (i, j), with i < j n, such that α i >α j. Let i(α) be the number of inversions of α.wedefine the alternant to be A x m P = α S m ( ) i(α) P ( x α(),...,x α(m) ), for any polynomial P(x,...,x n ). Definition 3 alphabet, i.e., Let be the operation of taking the Vandermonde determinant of an (X m ) = det ( x m j ) m i i, j=. Theorem (Sergeev s Formula) two alphabets. Then Let X m = x + +x m, and Y n = y + +y n be s λ [X m Y n ] = (X m ) (Y n ) Ax m Ay n X δ m m Y δ n n (i, j) λ (x i y j ) The notation (i, j) λ means that the point (i, j) belongs to the diagram of λ. Weset x i = 0 for i > m and y j = 0 for j > n. We use Sergeev s formula as a tool for making some calculations we need for the next two sections.. Let µ = ( e m ) be a hook. (We are assuming that e and m.) Let X ={x } and X ={x }. s µ [x x ] = ( ) e x m x e (x x ). (7). Let ν = (ν,ν ) be a two-row partition. Let Y ={y, y }. Then s ν [y + y ] = (y ( y ) ν y ν ν + y ν ν +). (8) y y 3. We say that a partition λ is a double hook if (, ) λ and it has the form λ = ( d d n 3 n 4 ). In particular any two-row shape is a double hook.

14 66 ROSAS Let λ be a double hook. Let U ={u, u } and V ={v,v }.Ifn 4 0 then s λ [u + u v v ] equals (u v )(u v )(u v )(u v ) ( ) d (u u ) n3 (v v ) d (u u )(v v ) ( u n 4 n 3 + u n 4 n 3 + )( v d + v d + ). (9) On the other hand, if n 4 = 0 then to compute s λ [u + u v v ] we should write λ as ( d d n 3 ). 4. Let λ be a hook shape, λ = ( d n ). (We are assuming that d and n.) Let U ={u, u } and V ={v,v }. Then s λ [u + u v v ] equals ( ) d (u u ) (v v ) { u v (u v )(u v )(u v )u n v d u v (u v )(u v )(u v )u n v d u v (u v )(u v )(u v )u n v d + u v (u v )(u v )(u v )u n v d. } (0) 5. The case of two hook shapes In this section we derive an explicit formula for the Kronecker coefficients γµν λ in the case in which µ = ( e u), and ν = ( f v) are both hook shapes. Given a partition λ the Kronecker coefficient γµν λ tells us whether point (u,v)belongs to some regions in N determined by µ, ν and λ. Lemma 3 Let (u,v) N and let R be the rectangle with vertices (a, b), (b, a), (c, d), and (d, c), with a b, c d, c a and d b. (Sometimes, when c = d = e, we denote this rectangle as (a, b; e).) Then (u,v) R if and only if v u a b and a + b u + v c + d Proof: Each of the four inequalities corresponds to whether the point (u,v) is in the proper half-plane formed by two of four edges of the rectangle. Theorem 3 Let λ, µ and ν be partitions of n, where µ = ( e u) and ν = ( f v) are hook shapes. Then the Kronecker coefficients γµν λ are given by the following:. If λ is a one-row shape, then γµν λ = δ µ,ν.. If λ is not contained in a double hook shape, then γµν λ = Let λ = ( d d n 3 n 4 ) be a double hook. Let x = d + d. Then (( γµν λ = n 3 e + f x n 4 ))(( f e d )) (( + n 3 e + f x + n 4 ))(( f e d + )).

15 KRONECKER PRODUCT OF SCHUR FUNCTIONS 67 Note that if n 4 = 0, then we shall rewrite λ = ( d d n 3 ) before using the previous formula. 4. Let λ = ( d w) be a hook shape. Suppose that e u, f v, and d w. Then γ λ µν = ((e d + f ))((d e + f ))(( f e + d)). Proof: Set X ={, x} and Y ={, y} in the comultiplication expansion () to obtain s λ [( x)( y)] = γµν λ s µ[ x]s ν [ y], () µ,ν We use Eq. (7) to replace s µ and s ν in the right hand side of (). Then we divide the resulting equation by ( x)( y) to get Therefore, s λ [ y x + xy] ( x)( y) = γµν λ ( x)e ( y) f. µ,ν γµν λ = [( x)e ( y) f ] s λ[ y x + xy], ( x)( y) when µ and ν are hook shapes. Case. If λ is not contained in any double hook, then the point (3, 3) is in λ, and by Sergeev s formula, s λ [ y x + xy] equals zero. Case. Let λ = ( d d n 3 n 4 ) be a double hook. Set u =, u = xy, v = x, and v = y in (9). Then we divide by ( x)( y) on both sides of the resulting equation to obtain s λ [ y x + xy] = ( ) d (xy) n 3+d ( x)( y) ( (xy) n 4 n 3 + )( x d + y d ) + ( x)( y). () xy x y Note: If n 4 = 0 then we should write λ = ( d d n 4 ) in order to use (9). Let ω p (T ) = (i, j) T ω p(i, j) be the generating function of a region T in N. Let R be the rectangle with vertices (0, d ), (d, 0), (d +n 4 n 3, n 4 n 3 ) and (n 4 n 3, d +n 4 n 3 ). Then ( (xy) n 4 n 3 + )( x d + y d ) + ω (d +n 4 n 3,n 4 n 3 )(R) = xy x y See figure 3. = n 4 n 3 k=0 i+ j=d (xy) k x i y j.

16 68 ROSAS Figure 3. d = 4 and n 3 n 4 = 4. The four vertices of R in Figure 3 are (0, 4), (4, 0), (8, 4), and (4, 8). We interpret the right-hand side of () as the sum of four different generating functions. To be more precise, the right-hand side of () can be written as 4 i= ω p i (r i ) where p = (n 4 + d, n 4 + d + d ) and R ={n 3 + d + d, n 3 + d ; n 4 n 3 }, p = (n 4 + d, n d + d ) and R ={n 3 + d + d, n 3 + d ; n 4 n 3 }, p 3 = (n 4 + d, n 4 + d + d ) and R 3 ={n 3 + d + d, n 3 + d ; n 4 n 3 }, and p 4 = (n 4 + d + d, n 4 + d + d ) and R 4 ={n 3 + d + d, n 3 + d + d ; n 4 n 3 }. We observe that R R (and R 3 R 4 ) are rectangles in N. Moreover, γ λ µν = (((e, f ) R R )) + (((e, f ) R 3 R 4 )). (3) The vertices of rectangle R R 4 are given (using the notation of Lemma 3) by a = n 3 + d + d b = n 3 + d c = n 4 + d + d d = n 4 + d Similarly, the vertices of rectangle R R 3 are given by a = n 3 + d + d b = n 3 + d c = n 4 + d + d d = n 4 + d Applying Lemma 3 to (3) we obtain (( γµν λ = n 3 e + f x n 4 ))(( f e d )) (( + n 3 e + f x + n 4 ))(( f e d + )). Case 3 (λ is a hook). Suppose that λ is a hook, λ = ( d w). Set u =, u = xy, v = x, and v = y in (0). Then we divide by ( x)( y) on both sides of the resulting

17 KRONECKER PRODUCT OF SCHUR FUNCTIONS 69 equation to obtain s λ [ y x + xy] ( x)( y) ( x = ( ) d d+ y d+ )( (xy) w ) x y xy ( x + ( ) d d y d )( (xy) w xy x y xy ). (4) We want to interpret this equation as a generating function for a region T using the weight ω. We proceed as follows: Let R be the rectangle with vertices (d, 0), (0, d), (d + w,w ), and (w, d + w ). Then ( (xy) w )( x d+ y d+ ) ω (w,d+w ) (R ) = = xy x y w k=0 i+ j=d (xy) k x i y j. (5) (See figure 3.) Similarly, let R be the rectangle with vertices (d, ), (, d), (d + w, w ), and (w, d + w ). Then ( (xy) w )( x d y d ) ω (w,d+w ) (R ) = xy xy x y w = xy (xy) k x i y j. (6) k=0 i+ j=d Observe that the points of N that can be reached from (0, d) in R and the points of N that can be reached from (, d) in R are disjoint. Moreover, they completely fill the rectangle R R. See figure 4. Note that R is contained in R. We obtain that ω (w,d+w ) (R ) + ω (w,d+w ) (R ) = (((e, f ) R )) Figure 4. d = 4,w = 6.

18 70 ROSAS We use apply Lemma 3 to the previous equation to obtain: (( e f d))((d e + f d + w )). But, by hypothesis, e u, f v, and d w. Therefore, this system is equivalent to ((d e + f ))(( f e + d))((e d + f )), as desired. Corollary 3 Let λ, µ, and ν be partitions of n, where µ = ( e u) and ν = ( f v) are hook shapes and λ = (λ,λ ) is a two-row shape. Then the Kronecker coefficients γµν λ are given by (( γµν λ = ((λ e λ ))((e = f )) + λ k e + f + λ ))(( e f )). Proof: In Theorem 3, set d = d = 0, n 3 = λ and n 4 = λ. Corollary 4 Let λ, µ and ν be partitions of n, where µ and ν are hook shapes. Then the Kronecker coefficients are bounded. Moreover, the only possible values for the Kronecker coefficients are 0, or. 6. The case of a hook shape and a two-row shape In this section we derive an explicit formula for the Kronecker coefficients in the case µ = ( e m ) is a hook and ν = (ν,ν ) is a two-row shape. Using the symmetry properties of the Kronecker product, we may assume that if λ = ( d d n 3 n 4 ) then n 4 n 3 d. (If n 4 = 0 then we should rewrite λ as ( d d n 3 ). Moreover, our hypothesis becomes n 3 d.) Theorem 4 Let λ, µand ν be partitions of n, where µ = ( e m ) is a hook and ν = (ν,ν ) is a two-row shape. Then the Kronecker coefficients γµν λ are given by the following:. If λ is a one-row shape, then γµν λ = δ µ,ν.. If λ is not contained in any double hook, then γµν λ = Suppose λ = ( d d n 3 n 4 ) is a double hook. Assume that n 4 n 3 d. (If n 4 = 0, then we should write λ = ( d d n 3 ).) Then γµν λ = ((n 3 ν d n 4 ))((d + d < e < d + d + 3)) + ((n 3 ν d n 4 ))((d + d e d + d + 3)) + ((n 3 ν d + n 4 ))((d + d < e < d + d + 3)) ((n 3 + d + d = ν ))((d + d + e d + d + )). 4. If λ is a hook, see Corollary 3. Proof: Set X = + x and Y = + y in the comultiplication expansion () to obtain s λ [( x)( + y)] = γµν λ s µ[ x]s ν [ + y]. (7) µ,ν

19 KRONECKER PRODUCT OF SCHUR FUNCTIONS 7 Use (7) and (8) to replace s µ and s ν in the right-hand side of (7), and divide by ( x) to obtain s λ [( x)( + y)] x = µ=( e m ) ν=(ν,ν ) γ λ µν ( x)e y ν ( y ν ν + y ). (8) If λ is not contained in any double hook, then the point (3, 3) is in λ, and by Sergeev s formula, s λ [( x)( + y)] equals zero. Since we already computed the Kronecker coefficients when λ is contained in a hook, we can assume for the rest of this proof that λ is a double hook. Let λ = ( d d n 3 n 4 ). (Note: If n 4 = 0 then we should write λ = ( d d n 4 ).) Set u =, u = y, v = x, and v = xy in (9), and multiply by y on both sides of x the resulting equation. µ=( e m ) ν=(ν,ν ) γ λ µν ( x)e y ν ( y ν ν + ) = (y x)( xy)( x) (( )( ( x) d +d y n 3+d y n 4 n 3 + y ) ) d +. (9) y We have that (y x)( xy)( x) = y x( + y + y ) + x ( + y + y ) x 3 y. Therefore, looking at the coefficient of x on both sides of the equation, we see that γ λ µν is zero if e is different from d + d, d + d +, d + d +, or d + d + 3. Let e = d + d or e = d + d + 3. Since ν n/, we have that γ λ µν = [yν ] = [y ν ] µ=( e m ) ν=(ν,ν ) µ=( e m ) ν=(ν,ν ) γ λ µν yν γ λ µν yν ( y ν ν + ) ( = [y ν ]y ) ( n 3+d y d + y n 4 n 3 + ) ( = [y ν ]y ) n 4 n 3 n 3+d y d + n 4 n 3 = [y ν ]y n 3+d k=0 y k. k=0 y y k (ν + > n/) (Eq. 9) (n 3 + d + d n/) We have obtained that for e = d + d or e = d + d + 3 γ λ µν = ((n 3 ν d n 4 )).

20 7 ROSAS Let e = d + d + ore = d + d +. Since ν n we have that γ λ µν = [yν ] = [y ν ] µ=( e m ) ν=(ν,ν ) µ=( e m ) ν=(ν,ν ) γ λ µν yν γµν λ ( y ν ν + ) = [y ν ]y n 3+d ( + y + y ) ( y d + ) ( y n 4 n 3 + ) y ( ( = [y ν ]y n 3+d y ( + y + y n 4 n 3 + )) ) ((n 3 + d + d = ν )) y ( n 4 n 3 ) = [y ν ]y n 3+d ( + y + y ) y k ((n 3 + d + d = ν )) We have obtained that for e = d + d + ore = d + d + k=0 γµν λ = ((n 3 ν d n 4 )) + ((n 3 ν d n 4 )) + ((n 3 ν d + n 4 )) ((n 3 + d + d = ν )). (30) Corollary 5 The Kronecker cofficients, γµν λ, where µ is a hook and ν is a two-row shape are always 0,, or Final comments The inner product of symmetric functions was discovered by J.H. Redfield [8] in 97, together with the scalar product of symmetric functions. He called them cup and cap products, respectively. D.E. Littlewood [5, 6] reinvented the inner product in 956. I.M. Gessel [3] and A. Lascoux [4] obtained combinatorial interpretations for the Kronecker coefficients in some restricted cases; Lascoux in the case where µ and ν are hooks, and λ a straight tableaux, and Gessel in the case that µ and ν are zigzag shapes and λ is an arbitrary skew shape. A.M. Garsia and J.B. Remmel [] founded a way to relate shuffles of permutations and Kronecker coefficients. From here they obtained a combinatorial interpretation for the Kronecker coefficients when λ is a product of homogeneous symmetric functions, and µ and ν are arbitrary skew shapes. They also showed how Gessel s and Lascoux s results are related. More recently, J.B. Remmel [9, 0], and J.B. Remmel and T. Whitehead [] have obtained formulas for computing the Kronecker coefficients in the same cases considered in this paper. Their approach was mainly combinatorial. They expanded the Kronecker product s µ s ν in terms of Schur functions using the Garsia-Remmel algorithm []. By doing this, the problem of computing the Kronecker coefficients was reduced to computing signed

21 KRONECKER PRODUCT OF SCHUR FUNCTIONS 73 sums of certain products of skew Schur functions. In general, it is not obvious how to go from the determination of the Kronecker coefficients γµ,ν λ when µ and ν are two-row shapes found in this paper, and the one obtained by J.B. Remmel and T. Whitehead []. But, in some particular cases this is easy to see. For instance, when λ is also a two-row shape, both formulas are exactly the same. Acknowledgments I would like to thank Ira Gessel for introducing me to this problem, and for his advise, Christiane Czech for pointing out some notational problems that were present in an earlier version, and the anonymous referee for many suggestions to improve the clarity of this article. References. N. Bergeron and A.M. Garsia, Sergeev s formula and the Littlewood-Richardson Rule, Linear and Multilinear Algebra 7 (990), A.M. Garsia and J.B. Remmel, Shuffles of permutations and Kronecker products, Graphs Combin. (3) (985), I.M. Gessel, Multipartite P-partitions and inner products of Schur functions, Contemp. Math. (984), A. Lascoux, Produit de Kronecker des representations du group symmetrique, Lecture Notes in Mathematics Springer Verlag, 795 (980), D.E. Littlewood, The Kronecker product of symmetric group representations, J. London Math. Soc. 3 (956), D.E. Littlewood, Plethysm and inner product of S-functions, J. London Math. Soc. 3 (957), I.G. Macdonald, Symmetric Functions and Hall Polynomials, second edition, Oxford: Oxford University Press, J.H. Redfield, The theory of group reduced distribution, Amer. J. Math. 49 (97), J.B. Remmel, A formula for the Kronecker product of Schur functions of hook shapes, J. Algebra 0 (989), J.B. Remmel, Formulas for the expansion of the Kronecker products S (m,n) S ( p r,r) and S ( k l ) S ( p r,r), Discrete Math. 99 (99), J.B. Remmel and T. Whitehead, On the Kronecker product of Schur functions of two row shapes, Bull. Belg. Math. Soc. Simon Stevin (994), M.H. Rosas, A combinatorial overview of the theory of MacMahon symmetric functions and a study of the Kronecker product of Schur functions, Ph.D. Thesis, Brandeis University (999). 3. B.E. Sagan, The Symmetric Group, Wadsworth & Brooks/Cole, Pacific Grove, California, A.N. Sergeev, The tensor algebra of the identity representation as a module over the Lie superalgebras gl(n, m) and Q(n), Math. USSR Sbornik, 5, pp