The Kronecker product of Schur functions indexed by two-row shapes or hook shapes

arXiv:math/0001084v1 [math.CO] 14 Jan 2000 THE KRONECKER PRODUCT OF SCHUR FUNCTIONS INDEXED BY TWO-ROW SHAPES OR HOOK SHAPES. MERCEDES H. ROSAS Abstract. The Kronecker product of two Schur functions sµand sν, denoted by sµ∗sν, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions µand ν. The coefficient of sλin this product is denoted by γλ µν , and corresponds to the multiplicity of the irreducible character χλin χµχν. We use Sergeev’s Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for sλ[XY ] to find closed formulas for the Kronecker coefficients γλ µν when λis an arbitrary shape and µand νare hook shapes or two-row shapes. Remmel [9, 10] and Remmel and Whitehead [11] 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. 1. 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, γλ µν, arise when expressing a Kronecker product (also called inner or internal product), sµ∗sν, of Schur functions in the Schur basis, sµ∗sν=X µ,ν γλ µν sλ.(1) 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={x1, x2,· · ·} and Y={y1, y2,· · · } sλ[XY ] = X µ,ν γλ µνsµ[X]sν[Y],(2) where sλ[XY ] means sλ(x1y1, x1y2,···, xiyj,···). Remmel [9, 10] and Remmel and Whitehead [11] have studied the Kronecker product of Schur functions corresponding to two two-row shapes, two hook shapes, and a hook shape and a 1 2 MERCEDES H. ROSAS two-row shape. We will use the comultiplication expansion (2) for the Kronecker coefficients, and a formula for expanding a Schur function of a difference of two alphabets due to Sergeev [1] 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 N2. 2. Basic definitions A partition λof a positive integer n, written as λ⊢n, is an unordered sequence of natural numbers adding to n. We write λas λ= (λ1, λ2,···, λn), where λ1≥ λ2≥ · · · , and consider two such strings equal if they differ by a string of zeroes. The nonzero numbers λiare 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= (1d12d2···ndn) for the partition of nthat has dicopies of i. Using this notation, we define the integer zλto be 1d1d1! 2d2d2!···ndndn!. We identify λwith the set of points (i, j) in N2defined by 1 ≤j≤λi, and refer to them as the Young diagram of λ. The Young diagram of a partition λis thought of as a collection of boxes arranged using matrix coordinates. For instance, the Young diagram corrresponding to λ= (4,3,1) is To any partition λwe associate the partition λ′, its conjugate partition, defined by λ′ i=|{j:λj≥i}| .Geometrically, λ′can be obtained from λby flipping the Young diagram of λaround its main diagonal. For instance, the conjugate partition of λ is λ′= (3,2,2,1), 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 [12] for proofs and details. Let R(Sn) be the space of class function in Sn, the symmetric group on nletters, and let Λnbe the space of homogeneous symmetric functions of degree n. A basis for R(Sn) is given by the characters of the irreducible representations of Sn. Let χµbe the irreducible character of Sncorresponding to the partition µ. There is a scalar product h,iSnon R(Sn) defined by hχµ, χνiSn=1 n!X σ∈Sn χµ(σ)χν(σ), KRONECKER PRODUCT 3 and extended by linearity. A basis for the space of symmetric functions is given by the Schur functions. There exists a scalar product h,iΛnon Λndefined by hsλ, sµiΛn=δλµ, where δλµ is the Kronecker delta, and extended by linearity. Let pµbe the power sum symmetric function corresponding to µ, where µis a partition of n. There is an isometry chn:R(Sn)7→ Λn, given by the characteristic map, chn(χ) = X µ⊢n z−1 µχ(µ)pµ. This map has the remarkable property that if χλis the irreducible character of Snindexed by λ, then chn(χλ) = sλ, the Schur function corresponding to λ. In particular, we obtain that sλ=Pµ⊢nz−1 µχλ(µ)pµ.Hence, χλ(µ) = hsλ, pµi.(3) Let λ,µ, and νbe partitions of n. The Kronecker coefficients γλ µν are defined by γλ µν =hχλ, χµχνiSn=1 n!X σ∈Sn χλ(σ)χµ(σ)χν(σ).(4) Equation (4) shows that the Kronecker coefficients γλ µν are symmetric in λ,µ, and ν. The relevance of the Kronecker coefficients comes from the following fact: Let Xµbe the representation of the symmetric group corresponding to the character χµ. Then χµχνis the character of Xµ⊗Xν, the representation obtained by taking the tensor product of Xµand Xν. Moreover, γλ µν is the multiplicity of Xλin Xµ⊗Xν. Let fand gbe homogeneous symmetric functions of degree n. The Kronecker product, f∗g, is defined by f∗g= chn(uv),(5) where chnu=f, chnv=g, and uv(σ) = u(σ)v(σ). To obtain (1) from this definition, we set f=sµ,g=sν,u=χµ, and v=χνin (5). The Kronecker product has the following symmetries: sµ∗sν=sν∗sµ. sµ∗sν=sµ′∗sν′. Moreover, if λis a one-row shape γλ µν =δµ,ν. 4 MERCEDES H. ROSAS We introduce the operation of substitution or plethysm into a symmetric function. Let fbe a symmetric function, and let X={x1, x2,· · · } be an alphabet. We write X=x1+x2+···, and define f[X] by, f[X] = f(x1, x2,···). In general, if uis any element of Q[[x1, x2,···]], we write uas Pαcαuαwhere uα is a monomial with coefficient 1. Then pλ[u] is defined by setting pn[u] = X α cαun α pλ[u] = pλ1[u]···pλn[u] for λ= (λ1,···, λn). We define f[u] for all symmetric functions fby saying that f[u] is linear in f. The operation of substitution into a symmetric function has the following properties. For αand βrational numbers, (αf +βg)[u] = αf[u] + βg[u].Moreover, if cα= 1 for all α, then f[u] = f(···, uα,···). Let X=x1+x2+··· and Y=y1+y2+··· be two alphabets. Define the sum of two alphabets by X+Y=x1+x2+···+y1+y2+· · · ,and the product of two alphabets by XY =x1y1+···+xiyj+···.Then pn[X+Y] = pn[X] + pn[Y], pn[XY ] = pn[X]pn[Y].(6) The inner product of function in the space of symmetric functions in two infinite alphabets is defined by h,iXY =h,iXh,iY, where for any given alphabet Z,h,iZdenotes the inner product of the space of symmetric functions in Z. For all partitions ρ, we have that pρ[XY ] = pρ[X]pρ[Y].If we rewrite (3) as pρ=Pλχλ(ρ)sλ, then X λ χλsλ[XY ] = X µ,ν χµχνsµ[X]sν[Y].(7) Taking the coefficient of χλon both sides of the previous equation we obtain sλ[XY ] = Xhχλ, χµχνisµ[X]sν[Y]. Finally, using the definition of Kronecker coefficients (4) we obtain the comultiplication expansion (2). KRONECKER PRODUCT 5 Notation. Let pbe a point in N2.We say that (i, j) can be reached from p, written p;(i, j), if (i, j) can be reached from pby moving any number of steps south-west or north-west. We define the weight function ωby ωp(i, j) = (xiyj,if p;(i, j), 0,otherwise. In particular, σk,l(h) = 0 if h < 0. Notation. We denote by ⌊x⌋the largest integer less than or equal to xand by ⌈x⌉the smallest integer greater than or equal to x. If fis a formal power series, then [xα]fdenotes the coefficient of xαin f. Following Donald Knuth we denote the characteristic function applied to a proposition Pby enclosing Pwith brackets, (P) = (1,if proposition Pis true, 0,otherwise. 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= (µ1, µ2) and ν= (ν1, ν2) are two-row shapes, and when we do not have any restriction on the partition λ. We describe the Kronecker coefficients γλ µν in terms of paths in N2. More precisely, we define two rectangular regions in N2using the parts of λ. Then we count the number of points in N2inside each of these rectangles that can be reached from (ν2, µ2+ 1), 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 4 in a concise form. Notation. We use the coordinate axes as if we were working with matrices with first entry (0,0). That is, the point (i, j) belongs to the ith row and the jth column. Lemma 1. Let kand lbe positive numbers. Let Rbe the rectangle with width k, height l, and upper–left square (0,0). Define σk,l(h) = {(u, v)∈R∩N2: (0, h);(u, v)} Then σk,l(h) =                0,if h < 0 ⌊(h 2+ 1)2⌋,if 0≤h < min(k, l) σk,l(s) + (h−s 2) min(k, l),if min(k, l)≤h < max(k, l) ⌈kl 2⌉ − σk,l(k+l−h−4),if his even and max(k, l)≤h ⌊kl 2⌋ − σk,l(k+l−h−4),if his odd and max(k, l)≤h 6 MERCEDES H. ROSAS where sis defined as follows: If h−min(k, l)is even, then s= min(k, l)−2; otherwise s= min(k, l)−1. Proof. If his to the left of the 0th column, then we cannot reach any of the points in N2inside R. Hence, σk,l(h) should be equal to zero. If 0 ≤h≤min(k, l), then we are counting the number of points in N2that can be reached from (0, h) inside the square Sof side min(k, l). We have to consider two cases. If his odd, then we are summing 2 + 4 + ···+ (h+ 1) = ⌊(h 2+ 1)2⌋. On the other hand, if his even, then we are summing 1 + 3 + ···+ (h+ 1) = (h 2+ 1)2. If min(k, l)≤h < max(k, l), then we subdivide our problem into two parts. First, we count the number of points in N2that can be reached from (0, h) inside the square Sby σk,l(s). Then we count those points in N2that are in Rbut not in S. Since h < max(k, l) all diagonals have length min(k, l) and there are h−s 2of them. See Table 1. If max(k, l)≤h, then it is easier to count the total number of points in N2that can be reached from (0, h) inside Rby choosing another parameter ˆ hbig enough and with the same parity as h. Then we subtract those points in N2in Rthat are not reachable from (0, h) because his too close. If his even this number is ⌈kl/2⌉. If his odd this number is ⌊kl/2⌋. Then we subtract those points that we should not have counted. we express this number in terms of the function σ. The line y=−x+h+ 2 intersects the line y=l−1 at x=h−l+ 3. This is the xcoordinate of the first point on the last row that is not reachable from (0, h). Then to obtain the number of points that can be reached from this point by moving south-west or north-west, we subtract h−l+ 3 to k−1. We have obtained that are σk,l(k+l−h−4) points that we should not have counted. Example 2. By definition σ9,5(4) counts the points in N2in Table 1 marked with ◦. Then σ9,5(4) = 9. Similarly, σ9,5(8) counts the points in N2in Table 1 marked either with the symbol ◦or with the symbol •. Then σ9,5(8) = 19. ◦◦◦•• ◦ ◦ • • ◦ ◦ • • ◦ • • ◦ • • Table 1. Lemma 3. Let a, b, c, and dbe in N. Let Rbe the rectangle with vertices (a, c), (a+b, c),(a, c +d), and (a+b, c +d). We define Γ(a, b, c, d)(x, y) = {(u, v)∈R: (x, y);(u, v)}. KRONECKER PRODUCT 7 Suppose that (x, y)is such that x≥y. Then Γ(a, b, c, d)(x, y) =      σb+1,d+1(x+y−a−c),0≤y≤c σb+1,y−c+1(x−a) + σb+1,c+d−y+1(x−a)−δ, c < y < c +d σb+1,d+1(x−y+c+d−a), c +d≤y where δis defined as follows If x < a, then δ= 0. If a≤x≤a+b, then δ=x−a+1 2. Finally, if x > a +bthen we consider two cases: If x−a−bis even then δ=b+1 2; otherwise, δ=b+1 2. Proof. We consider three cases. If 0 ≤y≤cthen the first position inside Rthat 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 Rthat 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, then we subdivide the problem in two parts. The number of position to the north of us is counted by σb+1,y−c+1(x−a). The number of position to the south of us is counted by σb+1,c+d−y+1(x−a). We define δ to be the number of points in N2that we counted twice during this process. Then it is easy to see that δis given by the previous definition. To compute the coefficient uνin the expansion f[X] = Pηuηsη[X] for f∈Λ, it is enough to expand f[x1+···+xn] = Pηuηsη[x1+···+xn] 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. Let µand νbe two-row partitions. Set X= 1 + xand Y= 1 + yin the comultiplication expansion (2) to obtain sλ[(1 + y)(1 + x)] = X µ,ν γλ µνsµ[1 + y]sν[1 + x].(8) Note that the Kronecker coefficients are zero when l(λ)>4. Jacobi’s definition of a Schur function on a finite alphabet sλ[X] as a quotient of alternants says that sλ[X] = sλ(x1,···, xn) = det(xλj+n−j i)1≤i,j≤n Qi<j(xi−xj).(9) By the symmetry properties of the Kronecker product it is enough to compute the Kronecker coefficients γλ µν when ν2≤µ2. Theorem 4. Let µ,ν, and λbe partitions of n, where µ= (µ1, µ2)and ν= (ν1, ν2) are two two-row partitions and let λ= (λ1, λ2, λ4, λ4)be a partition of length less than or equal to 4. Assume that ν2≤µ2. Then γλ µν =Γ(a, b, a +b+ 1, c)−Γ(a, b, a +b+c+d+ 2, c)(ν2, µ2+ 1). 8 MERCEDES H. ROSAS where a=λ3+λ4,b=λ2−λ3,c= min(λ1−λ2, λ3−λ4)and d=λ1+λ4−λ2−λ3. Proof. We expand the polynomial sλ[(1+y)(1+x)] = sλ(1, y, x, xy) in two different ways and obtain the Kronecker coefficients by equating both results. Let ϕbe the polynomial defined by ϕ= (1 −x)(1 −y)sλ(1, y, x, xy).Using Jacobi’s definition of a Schur function we obtain ϕ=  1 1 1 1 yλ1+3 yλ2+2 yλ3+1 yλ4 xλ1+3 xλ2+2 xλ3+1 xλ4 (xy)λ1+3 (xy)λ2+2 (xy)λ3+1 (xy)λ4  xy(1 −xy)(y−x)(1 −x)(1 −y).(10) On the other hand, we may use Jacobi’s definition to expand sµ[1+y] and sν[1+x] as quotients of alternants. Substitute this into (8): sλ[(1 + y)(1 + x)] = X µ=(µ1,µ2) ν=(ν1,ν2) γλ µνyµ2−yµ1+1 1−yxν2−xν1+1 1−x =X µ=(µ1,µ2) ν=(ν1,ν2) γλ µν xν2yµ2−xν2yµ1+1 −xν1+1yµ2+xν1+1yµ1+1 (1 −x)(1 −y).(11) Since ν1+ 1 and µ1+ 1 are both greater than ⌊n 2⌋,equation (11) implies that the coefficient of xν2yµ2in ϕis γλ µν.It is convenient to define an auxiliary polynomial by ζ= (1 −xy)(y−x)ϕ.(12) Let ξbe the polynomial obtained by expanding the determinant appearing in (10). Equations (10) and (12) imply ζ=ξ xy(1 −x)(1 −y). Let ξi,j be the coefficient of xiyjin ξ. ( Then ξi,j is zero if i≤0 or j≤0, because ξis a polynomial divisible by xy.) Let ζi,j be the coefficient of xiyjin ζ. Then X i,j≥0 ζi,jxiyj=1 xy(1 −x)(1 −y)X i,j≥0 ξi,jxiyj=X i,j,k,l≥0 ξi−k,j−lxi−1yj−1.(13) Comparing the coefficient of xiyjon both sides of equation (13) we obtain that ζi,j =X k,l≥0 ξi+1−k,j+1−l= i X k=0 j X l=0 ξk+1,l+1 (14) KRONECKER PRODUCT 9 We compute ζi,j from (14) by expanding the determinant appearing on (10). We consider two cases. Case 1. Suppose that λ1+λ4> λ2+λ3. Then λ1+λ2+ 4 > λ1+λ3+ 3 > λ1+λ4+ 2 ≥λ2+λ3+ 2 > λ2+λ4+ 1 > λ3+λ4. We record the values of ξj+1,i+1 in Table 2. We use the convention that ξi+1,j+1 is zero whenever the (i, j) entry is not in Table 2. i\jλ3+λ4λ2+λ4+ 1 λ2+λ3+ 2 λ1+λ4+ 2 λ1+λ3+ 3 λ1+λ2+ 4 λ3+λ40−1+1 +1 −1 0 λ2+λ4+ 1 +1 0 −1−1 0 +1 λ2+λ3+ 2 −1 +1 0 0 +1 −1 λ1+λ4+ 2 −1 +1 0 0 +1 −1 λ1+λ3+ 3 +1 0 −1−1 0 +1 λ1+λ2+ 4 0 −1 +1 +1 −1 0 Table 2 The values of ξj+1,i+1 when λ1+λ4≥λ2+λ3 Equation (14) shows that the value of ζi,j can be obtained by adding the entries northwest of the point (i, j). In Table 3 we record the values of ζi,j. i\j I1I2I3I4I5I6I7 I10000000 I20 0 −1 0 +1 0 0 I30 +1 0 0 0 −1 0 I40000000 I50−1 0 0 0 +1 0 I60 0 +1 0 −100 I70000000 Table 3 The values of ζi,j when λ1+λ4≥λ2+λ3 where I1= [0, λ3+λ4), I2= [λ3+λ4, λ2+λ4], I3= [λ2+λ4+ 1, λ2+λ3+ 1], I4= [λ2+λ3+ 2, λ1+λ4+ 1], I5= [λ1+λ4+ 2, λ1+λ3+ 2], I6= [λ1+λ3+ 3, λ1+λ2+ 3], I7= [λ1+λ2+ 4,∞). Case 2. Suppose that λ1+λ4≤λ2+λ3. Then λ1+λ2+ 4 > λ1+λ3+ 3 > λ2+λ3+ 2 > λ1+λ4+ 2 > λ2+λ4+ 1 > λ3+λ4. 16 MERCEDES H. ROSAS 1 1 1 1 1 1 1 1 1 1 11111 1 1 1 1 1 1 1 1 1 1 Table 5. d1= 4 and n3−n4= 4 Recall that we are using matrix coordinates, and that the upper-left corner has coordinates (0,0). The coordinates of the four vertices of Rin Table 5 are (0,4), (4,0), (8,4), and (4,8). We interpret the right-hand side of (22) as the sum of four different generating functions. To be more precise, the right-hand side of (22) can be written as P4 i=1 ωpi(ri) where p1= (n4+d2−1, n4+d2+d1−1) and R1={n3+d2+d1−1, n3+ d2−1; n4−n3},p2= (n4+d2, n34 + d2+d1−1) and R2={n3+d2+d1, n3+d2− 1; n4−n3},p3= (n4+d2−1, n4+d2+d1) and R3={n3+d2+d1−1, n3+d2;n4−n3}, and p4= (n4+d2+d1, n4+d2+d1) and R4={n3+d2+d1, n3+d2+d1;n4−n3}. We observe that R1∪R2(and R3∪R4) are rectangles in N2. Moreover, γλ µν = ((e, f)∈R1∪R2) + ((e, f)∈R3∪R4).(23) The vertices of rectangle R1∪R4are given (using the notation of 12) by a=n3+d2+d1−1b=n3+d2−1 c=n4+d2+d1d=n4+d2 Similarly, the vertices of rectangle R2∪R3are given by a=n3+d2+d1b=n3+d2−1 c=n4+d2+d1d=n4+d2−1 Applying Lemma 12 to (23) we obtain γλ µν = (n3−1≤e+f−x 2≤n4)(|f−e| ≤ d1) + (n3≤e+f−x+ 1 2≤n4)(|f−e| ≤ d1+ 1). KRONECKER PRODUCT 17 Case 3. λis a hook. Suppose that λis a hook, λ= (1dw). Set u1= 1, u2=xy,v1=x, and v2=yin (20). Then we divide by (1 −x)(1 −y) on both sides of the resulting equation to obtain (24) sλ[1 −y−x+xy] (1 −x)(1 −y)= (−1)dxd+1 −yd+1 x−y1−(xy)w 1−xy  + (−1)d−1xy xd−yd x−y1−(xy)w−1 1−xy . We want to interpret this equation as a generating function for a region Tusing the weight ω. We proceed as follows: Let R1be the rectangle with vertices (d, 0),(0, d),(d+w−1, w −1), and (w−1, d +w−1). Then ω(w−1,d+w−1)(R1) = 1−(xy)w 1−xy xd+1 −yd+1 x−y= w−1 X k=0 X i+j=d (xy)kxiyj.(25) (See Table 5.) Similarly, let R2be the rectangle with vertices (d, 1),(1, d),(d+ w−2, w −1), and (w−1, d +w−2). Then ω(w−1,d+w−2)(R2) = xy 1−(xy)w−1 1−xy xd−yd x−y=xy w−2 X k=0 X i+j=d−1 (xy)kxiyj. (26) Observe that the points in N2that can be reached from (0, d) in R1and the points in N2that can be reached from (1, d) in R2are disjoint. Moreover, they completely fill the rectangle R1∪R2. See Table 6. 1 1−1 1 1−1 1 −1 1 1−1 1 −1 1 −1 1 1−1 1 −1 1 −1 1 −1 1 1−1 1 −1 1 −1 1 −1 1 1−1 1 −1 1 −1 1 1−1 1 −1 1 1−1 1 1 Table 6 d= 4, w = 6. 18 MERCEDES H. ROSAS Note that R2is contained in R1. We obtain that ω(w−1,d+w−1)(R1) + ω(w−1,d+w−2)(R2) = |(e, f)∈R1| We use apply Lemma 12 to the previous equation to obtain: (|e−f| ≤ d)(d≤e+f≤d+ 2w−2). 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 14. Let λ,µ, and νbe partitions of n, where µ= (1eu)and ν= (1fv)are hook shapes and λ= (λ1, λ2)is a two-row shape. Then the Kronecker coefficients γλ µν are given by γλ µν = (λ2−1≤e≤λ1)(e=f) + (λ2≤e+f+ 1 2≤λ1)(|e−f| ≤ 1). Proof. In Theorem 13, set d1=d2= 0, n3=λ2and n4=λ1. Corollary 15. Let λ,µand νbe partitions of n, where µand νare hook shapes. Then the Kronecker coefficients are bounded. Moreover, the only possible values for the Kronecker coefficients are 0,1or 2. 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= (1e1m2) is a hook and ν= (ν1, ν2) is a two-row shape. Given a partition λ, the Kronecker coefficients γλ µν tell us whether the point (e1, ν2) belongs to some regions in N2determined by µ,νand λ. Using the symmetry properties of the Kronecker product, we may assume that if λ= (1d12d2n3n4) then n4−n3≤d1. (If n4= 0 then we should rewrite λas (1d12d2−12n3). Moreover, our hypothesis becomes n3−2≤d1.) Recall that we denote the value of the characteristic function at proposition P by (P). Theorem 16. Let λ,µand νbe partitions of n, where µ= (1e1m2)is a hook and ν= (ν1, ν2)is a two-row shape. Then the Kronecker coefficients γλ µν are given by the following: 1. If λis a one-row shape, then γλ µν =δµ,ν. 2. If λis not contained in any double hook, then γλ µν = 0. KRONECKER PRODUCT 19 3. Suppose λ= (1d12d2n3n4)is a double hook. Assume that n4−n3≤d1. (If n4= 0, then we should write λ= (1d12d2−12n3).) Then γλ µν = (n3≤ν2−d2−1≤n4)(d1+ 2d2< e1< d1+ 2d2+ 3) + (n3≤ν2−d2≤n4)(d1+ 2d2≤e1≤d1+ 2d2+ 3) + (n3≤ν2−d2+ 1 ≤n4)(d1+ 2d2< e1< d1+ 2d2+ 3) −(n3+d2+d1=ν2)(d1+ 2d2+ 1 ≤e1≤d1+ 2d2+ 2). 4. If λis a hook, see Corollary 14. Proof. Set X= 1+xand Y= 1+yin the comultiplication expansion (2) to obtain sλ[(1 −x)(1 + y)] = X µ,ν γλ µνsµ[1 −x]sν[1 + y].(27) Use (17) and (18) to replace sµand sνin the right-hand side of (27), and divide by (1 −x) to obtain sλ[(1 −x)(1 + y)] 1−x=X µ=(1e1m2) ν=(ν1,ν2) γλ µν(−x)e1yν21−yν1−ν2+1 1−y.(28) If λis not contained in any double hook, then the point (3,3) is in λ, and by Sergeev’s formula, sλ[(1 −x)(1 + y)] equals zero. Since we already computed the Kronecker coefficients when λis contained in a hook, we can assume for the rest of this proof that λis a double hook. Let λ= (1d12d2n3n4). (Note: If n4= 0 then we should write λ= (1d12d2−12n4).) Set u1= 1, u2=y,v1=x, and v2=xy in (19), and multiply by 1−y 1−xon both sides of the resulting equation. (29) X µ=(1e1m2) ν=(ν1,ν2) γλ µν(−x)e1yν21−yν1−ν2+1= (y−x)(1 −xy)(1 −x) ×(−x)d1+2d2yn3+d2−1(1 −yn4−n3+1)(1 −yd1+1) 1−y. We have that (y−x)(1 −xy)(1 −x) = y−x(1 + y+y2) + x2(1 + y+y2)−x3y. Therefore, looking at the coefficient of xon both sides of the equation, we see that γλ µν is zero if e1is different from d1+ 2d2, d1+ 2d2+ 1, d1+ 2d2+ 2,or d1+ 2d2+ 3. 20 MERCEDES H. ROSAS Let e1=d1+ 2d2or e1=d1+ 2d2+ 3. Since ν2≤n/2, we have that γλ µν = [yν2]X µ=(1e1m2) ν=(ν1,ν2) γλ µνyν2 = [yν2]X µ=(1e1m2) ν=(ν1,ν2) γλ µνyν2(1 −yν1−ν2+1) (ν1+ 1 > n/2) = [yν2]yn3+d2(1 −yd1+1)1−yn4−n3+1 1−y(Eq. 29) = [yν2]yn3+d2(1 −yd1+1) n4−n3 X k=0 yk = [yν2]yn3+d2 n4−n3 X k=0 yk.(n3+d2+d1≥n/2) We have obtained that for e1=d1+ 2d2or e1=d1+ 2d2+ 3 γλ µν = (n3≤ν2−d2≤n4). Let e1=d1+ 2d2+ 1 or e1=d1+ 2d2+ 2. Since ν2≤ ⌊n 2⌋we have that γλ µν = [yν2]X µ=(1e1m2) ν=(ν1,ν2) γλ µνyν2 = [yν2]X µ=(1e1m2) ν=(ν1,ν2) γλ µν(1 −yν1−ν2+1) = [yν2]yn3+d2−1(1 + y+y2)(1 −yd1+1)1−yn4−n3+1 1−y =[yν2]yn3+d2−1(1 + y+y2)1−yn4−n3+1 1−y−(n3+d2+d1=ν2) =[yν2]yn3+d2−1(1 + y+y2) n4−n3 X k=0 yk−(n3+d2+d1=ν2) We have obtained that for e1=d1+ 2d2+ 1 or e1=d1+ 2d2+ 2 (30) γλ µν = (n3≤ν2−d2−1≤n4) + (n3≤ν2−d2≤n4) + (n3≤ν2−d2+ 1 ≤n4)−(n3+d2+d1=ν2). KRONECKER PRODUCT 21 Corollary 17. The Kronecker cofficients, γλ µν, where µis a hook and νis a tworow shape are always 0,1,2or 3. 7. Final comments The inner product of symmetric functions was discovered by J. H. Redfield [8] in 1927, 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 1956. More recently, I.M. Gessel [3] and A. Lascoux [4] obtained combinatorial interpretations for the Kronecker coefficients in some restricted cases; Lascoux in the case where µand νare hooks, and λa straight tableaux, and Gessel in the case that µand νare zigzag shapes and λis an arbitrary skew shape. A. Lascoux interpreted the Kronecker coefficients, when two of the shapes are hooks as counting clases of words under some equivalence relation. We refer to [4] or [2] for a complete statement of his results. The Corollary of Theorem 3 in this paper shows that each class of words, under Lascoux’s equivalence, contains either 0, 1, or 2 different representatives. I. Gessel worked on a more general framework, contemplating the occurrence of skew tableaux. It was shown in [2] that in the case where two of the partitions are hook shapes, and the third one is an arbitrary straight shape, his result is equivalent to Lascoux’s. In [2], 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 λis a product of homogeneous symmetric functions, and µand νare arbitrary skew shapes. They also showed how Gessel’s and Lascoux’s results are related. J.B. Remmel [9], [10], and J.B. Remmel and T. Whitehead [11], obtained formulas for computing the Kronecker coefficients in the same cases considered in this paper. Their approach was mainly combinatorial: First, they expanded the Kronecker product sµ∗sνin terms of Schur functions using the Garsia-Remmel algorithm [2]. The problem of computing the Kronecker coefficients was reduced to computing signed sums of certain products of skew Schur functions. Then they obtained a description of the coefficients that arise in the expansion of the resulting product of skew Schur functions in terms of counting 3-colored diagrams in [9], and [10] or 4-colored diagrams in [11]. At this point, they reduced the problem to computing a signed sum of colored diagram. Finally, they defined involutions on these signed sums to cancel negative terms, and obtained the desired formulas by counting classes of restricted colored diagram. In general, it is not obvious how to go from the determination of the Kronecker coefficients γλ µ,ν when µand νare two-row shapes found in this paper, and the one 22 MERCEDES H. ROSAS obtained by J.B. Remmel and T. Whitehead [11]. But, in some particular cases this is easy to see. For instance, when λis also a two-row shape, both formulas are exactly the same. References [1] N. Bergeron and A.M. Garsia, “Sergeev’s Formula and the Littlewood-Richardson Rule,” Linear and Multilinear Algebra 27, 1990, pp. 79–100. [2] A.M. Garsia and J.B. Remmel, “Shuffles of permutations and Kronecker products,” Graphs Combin. 1985, pp. 217–263. [3] I.M. Gessel, “Multipartite P-partitions and inner products of Schur functions,” Contemp. Math. 1984, pp. 289–302. [4] A. Lascoux, “Produit de Kronecker des representations du group symmetrique,” Lecture Notes in Mathematics 1980, 795, Springer Verlag pp. 319–329. [5] D.E. Littlewood, “The Kronecker product of symmetric group representations,” J. London Math. Soc. 31, 1956, pp. 89–93. [6] D.E. Littlewood, “Plethysm and inner product of S-functions,” J. London Math. Soc. 32, 1957, pp. 18–22. [7] I.G. Macdonald, Symmetric Functions and Hall Polynomials, second edition, Oxford University Press, 1995. [8] J.H. Redfield “The theory of group reduced distribution,” Amer. J. Math. 49, 1927, pp. 433-455. [9] J.B. Remmel, “A formula for the Kronecker product of Schur functions of hook shapes,” J. Algebra 120, 1989, pp. 100–118. [10] J.B. Remmel, “Formulas for the expansion of the Kronecker products S(m,n)⊗S(1p−r,r)and S(1k2l)⊗S(1p−r,r),” Discrete Math. 99, 1992, pp. 265–287. [11] J.B. Remmel and T. Whitehead, “On the Kronecker product of Schur functions of two row shapes,” Bull. Belg. Math. Soc. Simon Stevin 1, 1994, pp. 649–683. [12] B.E. Sagan, The Symmetric Group, Wadsworth & Brooks/Cole, Pacific Grove, California, 1991. Department of Mathematics, Brandeis University, Waltham, MA 02254 E-mail address:[email protected]s.edu