a Xi :ma h/0001084 1 [ma h.CO] 14 Jan 2000
THE KRONECKER PRODUCT OF SCHUR FUNCTIONS
INDEXED BY TWO-ROW SHAPES OR HOOK SHAPES.
MERCEDES H. ROSAS
Abs ac . The K onecke p oduc o wo Schu unc ions sµand sν, deno ed
by sµ∗sν, is he F obenius cha ac e is ic o he enso p oduc o he i educible
ep esen a ions o he symme ic g oup co esponding o he pa i ions µand ν.
The coe icien o sλin his p oduc is deno ed by γλ
µν , and co esponds o he
mul iplici y o he i educible cha ac e χλin χµχν.
We use Se gee ’s Fo mula o a Schu unc ion o a di e ence o wo alphabe s
and he comul iplica ion expansion o sλ[XY ] o ind closed o mulas o he
K onecke coe icien s γλ
µν when λis an a bi a y shape and µand νa e hook
shapes o wo- ow shapes.
Remmel [9, 10] and Remmel and Whi ehead [11] de i ed some closed o mulas
o he K onecke p oduc o Schu unc ions indexed by wo- ow shapes o hook
shapes using a di e en app oach. We belie e ha he app oach o his pape
is mo e na u al. The o mulas ob ained a e simple and e lec he symme y o
he K onecke p oduc .
1. In oduc ion
The aim o his pape is o de i e an explici o mula o he K onecke coe -
icien s co esponding o pa i ions o ce ain shapes. The K onecke coe icien s,
γλ
µν, a ise when exp essing a K onecke p oduc (also called inne o in e nal p od-
uc ), sµ∗sν, o Schu unc ions in he Schu basis,
sµ∗sν=X
µ,ν
γλ
µν sλ.(1)
These coe icien s can also be de ined as he mul iplici ies o he i educible ep e-
sen a ions in he enso p oduc o wo i educible ep esen a ions o he symme ic
g oup. A hi d way o de ine hem is by he comul iplica ion expansion. Gi en
wo alphabe s X={x1, x2,· · ·} and Y={y1, y2,· · · }
sλ[XY ] = X
µ,ν
γλ
µνsµ[X]sν[Y],(2)
whe e sλ[XY ] means sλ(x1y1, x1y2,···, xiyj,···). Remmel [9, 10] and Remmel
and Whi ehead [11] ha e s udied he K onecke p oduc o Schu unc ions co -
esponding o wo wo- ow shapes, wo hook shapes, and a hook shape and a
1
2 MERCEDES H. ROSAS
wo- ow shape. We will use he comul iplica ion expansion (2) o he K onecke
coe icien s, and a o mula o expanding a Schu unc ion o a di e ence o wo
alphabe s due o Se gee [1] o ob ain simila esul s in a simple way. We belie e
ha he o mulas ob ained using his app oach a e elegan and e lec he symme-
y o he K onecke p oduc . In he h ee cases we ound a way o exp ess he
K onecke coe icien s in e ms o egions and pa hs in N2.
2. Basic de ini ions
A pa i ion λo a posi i e in ege n, w i en as λ⊢n, is an uno de ed sequence
o na u al numbe s adding o n. We w i e λas λ= (λ1, λ2,···, λn), whe e λ1≥
λ2≥ · · · , and conside wo such s ings equal i hey di e by a s ing o ze oes.
The nonze o numbe s λia e called he pa s o λ, and he numbe o pa s is
called he leng h o λ, deno ed by l(λ). In some cases, i is con enien o w i e
λ= (1d12d2···ndn) o he pa i ion o n ha has dicopies o i. Using his no a ion,
we de ine he in ege zλ o be 1d1d1! 2d2d2!···ndndn!.
We iden i y λwi h he se o poin s (i, j) in N2de ined by 1 ≤j≤λi, and e e
o hem as he Young diag am o λ. The Young diag am o a pa i ion λis hough
o as a collec ion o boxes a anged using ma ix coo dina es. Fo ins ance, he
Young diag am co esponding o λ= (4,3,1) is
To any pa i ion λwe associa e he pa i ion λ′, i s conjuga e pa i ion, de ined by
λ′
i=|{j:λj≥i}| .Geome ically, λ′can be ob ained om λby lipping he Young
diag am o λa ound i s main diagonal. Fo ins ance, he conjuga e pa i ion o λ
is λ′= (3,2,2,1), and he co esponding Young diag am is
We ecall some ac s abou he heo y o ep esen a ions o he symme ic g oup,
and abou symme ic unc ions. See [7] o [12] o p oo s and de ails.
Le R(Sn) be he space o class unc ion in Sn, he symme ic g oup on nle e s,
and le Λnbe he space o homogeneous symme ic unc ions o deg ee n. A basis
o R(Sn) is gi en by he cha ac e s o he i educible ep esen a ions o Sn. Le
χµbe he i educible cha ac e o Snco esponding o he pa i ion µ. The e is a
scala p oduc h,iSnon R(Sn) de ined by
hχµ, χνiSn=1
n!X
σ∈Sn
χµ(σ)χν(σ),
KRONECKER PRODUCT 3
and ex ended by linea i y.
A basis o he space o symme ic unc ions is gi en by he Schu unc ions.
The e exis s a scala p oduc h,iΛnon Λnde ined by
hsλ, sµiΛn=δλµ,
whe e δλµ is he K onecke del a, and ex ended by linea i y.
Le pµbe he powe sum symme ic unc ion co esponding o µ, whe e µis a
pa i ion o n. The e is an isome y chn:R(Sn)7→ Λn, gi en by he cha ac e is ic
map,
chn(χ) = X
µ⊢n
z−1
µχ(µ)pµ.
This map has he ema kable p ope y ha i χλis he i educible cha ac e o
Snindexed by λ, hen chn(χλ) = sλ, he Schu unc ion co esponding o λ. In
pa icula , we ob ain ha sλ=Pµ⊢nz−1
µχλ(µ)pµ.Hence,
χλ(µ) = hsλ, pµi.(3)
Le λ,µ, and νbe pa i ions o n. The K onecke coe icien s γλ
µν a e de ined by
γλ
µν =hχλ, χµχνiSn=1
n!X
σ∈Sn
χλ(σ)χµ(σ)χν(σ).(4)
Equa ion (4) shows ha he K onecke coe icien s γλ
µν a e symme ic in λ,µ, and
ν. The ele ance o he K onecke coe icien s comes om he ollowing ac : Le
Xµbe he ep esen a ion o he symme ic g oup co esponding o he cha ac e
χµ. Then χµχνis he cha ac e o Xµ⊗Xν, he ep esen a ion ob ained by aking
he enso p oduc o Xµand Xν. Mo eo e , γλ
µν is he mul iplici y o Xλin
Xµ⊗Xν.
Le and gbe homogeneous symme ic unc ions o deg ee n. The K onecke
p oduc , ∗g, is de ined by
∗g= chn(u ),(5)
whe e chnu= , chn =g, and u (σ) = u(σ) (σ). To ob ain (1) om his
de ini ion, we se =sµ,g=sν,u=χµ, and =χνin (5).
The K onecke p oduc has he ollowing symme ies:
sµ∗sν=sν∗sµ.
sµ∗sν=sµ′∗sν′.
Mo eo e , i λis a one- ow shape
γλ
µν =δµ,ν.
4 MERCEDES H. ROSAS
We in oduce he ope a ion o subs i u ion o ple hysm in o a symme ic unc ion.
Le be a symme ic unc ion, and le X={x1, x2,· · · } be an alphabe . We w i e
X=x1+x2+···, and de ine [X] by,
[X] = (x1, x2,···).
In gene al, i uis any elemen o Q[[x1, x2,···]], we w i e uas Pαcαuαwhe e uα
is a monomial wi h coe icien 1. Then pλ[u] is de ined by se ing
pn[u] = X
α
cαun
α
pλ[u] = pλ1[u]···pλn[u]
o λ= (λ1,···, λn). We de ine [u] o all symme ic unc ions by saying ha
[u] is linea in .
The ope a ion o subs i u ion in o a symme ic unc ion has he ollowing p op-
e ies. Fo αand β a ional numbe s, (α +βg)[u] = α [u] + βg[u].Mo eo e , i
cα= 1 o all α, hen [u] = (···, uα,···).
Le X=x1+x2+··· and Y=y1+y2+··· be wo alphabe s. De ine he sum
o wo alphabe s by X+Y=x1+x2+···+y1+y2+· · · ,and he p oduc o wo
alphabe s by XY =x1y1+···+xiyj+···.Then
pn[X+Y] = pn[X] + pn[Y],
pn[XY ] = pn[X]pn[Y].(6)
The inne p oduc o unc ion in he space o symme ic unc ions in wo in ini e
alphabe s is de ined by
h,iXY =h,iXh,iY,
whe e o any gi en alphabe Z,h,iZdeno es he inne p oduc o he space o
symme ic unc ions in Z.
Fo all pa i ions ρ, we ha e ha pρ[XY ] = pρ[X]pρ[Y].I we ew i e (3) as
pρ=Pλχλ(ρ)sλ, hen
X
λ
χλsλ[XY ] = X
µ,ν
χµχνsµ[X]sν[Y].(7)
Taking he coe icien o χλon bo h sides o he p e ious equa ion we ob ain
sλ[XY ] = Xhχλ, χµχνisµ[X]sν[Y].
Finally, using he de ini ion o K onecke coe icien s (4) we ob ain he comul ipli-
ca ion expansion (2).
KRONECKER PRODUCT 5
No a ion. Le pbe a poin in N2.We say ha (i, j) can be eached om p,
w i en p;(i, j), i (i, j) can be eached om pby mo ing any numbe o s eps
sou h-wes o no h-wes . We de ine he weigh unc ion ωby
ωp(i, j) = (xiyj,i p;(i, j),
0,o he wise.
In pa icula , σk,l(h) = 0 i h < 0.
No a ion. We deno e by ⌊x⌋ he la ges in ege less han o equal o xand by
⌈x⌉ he smalles in ege g ea e han o equal o x.
I is a o mal powe se ies, hen [xα] deno es he coe icien o xαin .
Following Donald Knu h we deno e he cha ac e is ic unc ion applied o a
p oposi ion Pby enclosing Pwi h b acke s,
(P) = (1,i p oposi ion Pis ue,
0,o he wise.
3. The case o wo wo- ow shapes
The objec o his sec ion is o ind a closed o mula o he K onecke coe icien s
when µ= (µ1, µ2) and ν= (ν1, ν2) a e wo- ow shapes, and when we do no ha e
any es ic ion on he pa i ion λ. We desc ibe he K onecke coe icien s γλ
µν in
e ms o pa hs in N2. Mo e p ecisely, we de ine wo ec angula egions in N2using
he pa s o λ. Then we coun he numbe o poin s in N2inside each o hese
ec angles ha can be eached om (ν2, µ2+ 1), i we a e allowed o mo e any
numbe o s eps sou h-wes o no h-wes . Finally, we sub ac hese wo numbe s.
We begin by in oducing wo lemmas ha allow us o s a e Theo em 4 in a
concise o m.
No a ion. We use he coo dina e axes as i we we e wo king wi h ma ices wi h
i s en y (0,0). Tha is, he poin (i, j) belongs o he i h ow and he j h column.
Lemma 1. Le kand lbe posi i e numbe s. Le Rbe he ec angle wi h wid h k,
heigh l, and uppe –le squa e (0,0). De ine
σk,l(h) = {(u, )∈R∩N2: (0, h);(u, )}
Then
σk,l(h) =















0,i h < 0
⌊(h
2+ 1)2⌋,i 0≤h < min(k, l)
σk,l(s) + (h−s
2) min(k, l),i min(k, l)≤h < max(k, l)
⌈kl
2⌉ − σk,l(k+l−h−4),i his e en and max(k, l)≤h
⌊kl
2⌋ − σk,l(k+l−h−4),i his odd and max(k, l)≤h

6 MERCEDES H. ROSAS
whe e sis de ined as ollows: I h−min(k, l)is e en, hen s= min(k, l)−2;
o he wise s= min(k, l)−1.
P oo . I his o he le o he 0 h column, hen we canno each any o he poin s
in N2inside R. Hence, σk,l(h) should be equal o ze o.
I 0 ≤h≤min(k, l), hen we a e coun ing he numbe o poin s in N2 ha can
be eached om (0, h) inside he squa e So side min(k, l). We ha e o conside
wo cases. I his odd, hen we a e summing 2 + 4 + ···+ (h+ 1) = ⌊(h
2+ 1)2⌋. On
he o he hand, i his e en, hen we a e summing 1 + 3 + ···+ (h+ 1) = (h
2+ 1)2.
I min(k, l)≤h < max(k, l), hen we subdi ide ou p oblem in o wo pa s.
Fi s , we coun he numbe o poin s in N2 ha can be eached om (0, h) inside
he squa e Sby σk,l(s). Then we coun hose poin s in N2 ha a e in Rbu no
in S. Since h < max(k, l) all diagonals ha e leng h min(k, l) and he e a e h−s
2o
hem. See Table 1.
I max(k, l)≤h, hen i is easie o coun he o al numbe o poin s in N2 ha
can be eached om (0, h) inside Rby choosing ano he pa ame e ˆ
hbig enough
and wi h he same pa i y as h. Then we sub ac hose poin s in N2in R ha a e
no eachable om (0, h) because his oo close. I his e en his numbe is ⌈kl/2⌉.
I his odd his numbe is ⌊kl/2⌋.
Then we sub ac hose poin s ha we should no ha e coun ed. we exp ess his
numbe in e ms o he unc ion σ. The line y=−x+h+ 2 in e sec s he line
y=l−1 a x=h−l+ 3. This is he xcoo dina e o he i s poin on he las
ow ha is no eachable om (0, h). Then o ob ain he numbe o poin s ha
can be eached om his poin by mo ing sou h-wes o no h-wes , we sub ac
h−l+ 3 o k−1. We ha e ob ained ha a e σk,l(k+l−h−4) poin s ha we
should no ha e coun ed.
Example 2. By de ini ion σ9,5(4) coun s he poin s in N2in Table 1 ma ked wi h
◦. Then σ9,5(4) = 9. Simila ly, σ9,5(8) coun s he poin s in N2in Table 1 ma ked
ei he wi h he symbol ◦o wi h he symbol •. Then σ9,5(8) = 19.
◦◦◦••
◦ ◦ • •
◦ ◦ • •
◦ • •
◦ • •
Table 1.
Lemma 3. Le a, b, c, and dbe in N. Le Rbe he ec angle wi h e ices (a, c),
(a+b, c),(a, c +d), and (a+b, c +d). We de ine
Γ(a, b, c, d)(x, y) = {(u, )∈R: (x, y);(u, )}.
KRONECKER PRODUCT 7
Suppose ha (x, y)is such ha 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
whe e δis de ined as ollows I x < a, hen δ= 0. I a≤x≤a+b, hen
δ=x−a+1
2. Finally, i x > a +b hen we conside wo cases: I x−a−bis e en
hen δ=b+1
2; o he wise, δ=b+1
2.
P oo . We conside h ee cases. I 0 ≤y≤c hen he i s posi ion inside R ha
we each is (x+y−a−c, c). The e o e, we assume ha we a e s a ing a his
poin . Simila ly, i y≥c+d, hen he i s posi ion inside R ha we each is
(x−y+c+d−a, c). Again, we can assume ha we a e s a ing a his poin .
On he o he hand, i c < y < c+d, hen we subdi ide he p oblem in wo pa s.
The numbe o posi ion o he no h o us is coun ed by σb+1,y−c+1(x−a). The
numbe o posi ion o he sou h o us is coun ed by σb+1,c+d−y+1(x−a). We de ine δ
o be he numbe o poin s in N2 ha we coun ed wice du ing his p ocess. Then
i is easy o see ha δis gi en by he p e ious de ini ion.
To compu e he coe icien uνin he expansion [X] = Pηuηsη[X] o ∈Λ,
i is enough o expand [x1+···+xn] = Pηuηsη[x1+···+xn] o any n≥l(ν).
(See [7, sec ion I.3], o p oo s and de ails.) The e o e, in his sec ion we wo k wi h
symme ic unc ions in a ini e numbe o a iables.
Le µand νbe wo- ow pa i ions. Se X= 1 + xand Y= 1 + yin he
comul iplica ion expansion (2) o ob ain
sλ[(1 + y)(1 + x)] = X
µ,ν
γλ
µνsµ[1 + y]sν[1 + x].(8)
No e ha he K onecke coe icien s a e ze o when l(λ)>4.
Jacobi’s de ini ion o a Schu unc ion on a ini e alphabe sλ[X] as a quo ien
o al e nan s says ha
sλ[X] = sλ(x1,···, xn) = de (xλj+n−j
i)1≤i,j≤n
Qi<j(xi−xj).(9)
By he symme y p ope ies o he K onecke p oduc i is enough o compu e
he K onecke coe icien s γλ
µν when ν2≤µ2.
Theo em 4. Le µ,ν, and λbe pa i ions o n, whe e µ= (µ1, µ2)and ν= (ν1, ν2)
a e wo wo- ow pa i ions and le λ= (λ1, λ2, λ4, λ4)be a pa i ion o leng h less
han o equal o 4. Assume ha 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
whe e a=λ3+λ4,b=λ2−λ3,c= min(λ1−λ2, λ3−λ4)and d=λ1+λ4−λ2−λ3.
P oo . We expand he polynomial sλ[(1+y)(1+x)] = sλ(1, y, x, xy) in wo di e en
ways and ob ain he K onecke coe icien s by equa ing bo h esul s. Le ϕbe he
polynomial de ined by ϕ= (1 −x)(1 −y)sλ(1, y, x, xy).Using Jacobi’s de ini ion
o a Schu unc ion we ob ain
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 he o he hand, we may use Jacobi’s de ini ion o expand sµ[1+y] and sν[1+x]
as quo ien s o al e nan s. Subs i u e his in o (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 a e bo h g ea e han ⌊n
2⌋,equa ion (11) implies ha he
coe icien o xν2yµ2in ϕis γλ
µν.I is con enien o de ine an auxilia y polynomial
by
ζ= (1 −xy)(y−x)ϕ.(12)
Le ξbe he polynomial ob ained by expanding he de e minan appea ing in (10).
Equa ions (10) and (12) imply
ζ=ξ
xy(1 −x)(1 −y).
Le ξi,j be he coe icien o xiyjin ξ. ( Then ξi,j is ze o i i≤0 o j≤0, because
ξis a polynomial di isible by xy.) Le ζi,j be he coe icien o 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)
Compa ing he coe icien o xiyjon bo h sides o equa ion (13) we ob ain ha
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 compu e ζi,j om (14) by expanding he de e minan appea ing on (10). We
conside wo cases.
Case 1. Suppose ha λ1+λ4> λ2+λ3. Then
λ1+λ2+ 4 > λ1+λ3+ 3 > λ1+λ4+ 2 ≥λ2+λ3+ 2 > λ2+λ4+ 1 > λ3+λ4.
We eco d he alues o ξj+1,i+1 in Table 2. We use he con en ion ha ξi+1,j+1 is
ze o whene e he (i, j) en y is no 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 alues o ξj+1,i+1 when λ1+λ4≥λ2+λ3
Equa ion (14) shows ha he alue o ζi,j can be ob ained by adding he en ies
no hwes o he poin (i, j). In Table 3 we eco d he alues o 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 alues o ζi,j when λ1+λ4≥λ2+λ3
whe e
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 ha 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 ha we a e using ma ix coo dina es, and ha he uppe -le co ne has
coo dina es (0,0). The coo dina es o he ou e ices o Rin Table 5 a e (0,4),
(4,0), (8,4), and (4,8).
We in e p e he igh -hand side o (22) as he sum o ou di e en gene a -
ing unc ions. To be mo e p ecise, he igh -hand side o (22) can be w i en as
P4
i=1 ωpi( i) whe e 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 obse e ha R1∪R2(and R3∪R4) a e ec angles in N2. Mo eo e ,
γλ
µν = ((e, )∈R1∪R2) + ((e, )∈R3∪R4).(23)
The e ices o ec angle R1∪R4a e gi en (using he no a ion o 12) by
a=n3+d2+d1−1b=n3+d2−1
c=n4+d2+d1d=n4+d2
Simila ly, he e ices o ec angle R2∪R3a e gi en by
a=n3+d2+d1b=n3+d2−1
c=n4+d2+d1d=n4+d2−1
Applying Lemma 12 o (23) we ob ain
γλ
µν = (n3−1≤e+ −x
2≤n4)(| −e| ≤ d1)
+ (n3≤e+ −x+ 1
2≤n4)(| −e| ≤ d1+ 1).

KRONECKER PRODUCT 17
Case 3. λis a hook. Suppose ha λis a hook, λ= (1dw). Se u1= 1,
u2=xy, 1=x, and 2=yin (20). Then we di ide by (1 −x)(1 −y) on bo h
sides o he esul ing equa ion o ob ain
(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 wan o in e p e his equa ion as a gene a ing unc ion o a egion Tusing
he weigh ω. We p oceed as ollows:
Le R1be he ec angle wi h e ices (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.) Simila ly, le R2be he ec angle wi h e ices (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)
Obse e ha he poin s in N2 ha can be eached om (0, d) in R1and he
poin s in N2 ha can be eached om (1, d) in R2a e disjoin . Mo eo e , hey
comple ely ill he ec angle 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
No e ha R2is con ained in R1. We ob ain ha
ω(w−1,d+w−1)(R1) + ω(w−1,d+w−2)(R2) = |(e, )∈R1|
We use apply Lemma 12 o he p e ious equa ion o ob ain:
(|e− | ≤ d)(d≤e+ ≤d+ 2w−2).
Bu , by hypo hesis, e≤u, ≤ , and d≤w. The e o e, his sys em is equi alen
o (d≤e+ )( ≤e+d)(e≤d+ ), as desi ed.
Co olla y 14. Le λ,µ, and νbe pa i ions o n, whe e µ= (1eu)and ν=
(1 )a e hook shapes and λ= (λ1, λ2)is a wo- ow shape. Then he K onecke
coe icien s γλ
µν a e gi en by
γλ
µν = (λ2−1≤e≤λ1)(e= ) + (λ2≤e+ + 1
2≤λ1)(|e− | ≤ 1).
P oo . In Theo em 13, se d1=d2= 0, n3=λ2and n4=λ1.
Co olla y 15. Le λ,µand νbe pa i ions o n, whe e µand νa e hook shapes.
Then he K onecke coe icien s a e bounded. Mo eo e , he only possible alues
o he K onecke coe icien s a e 0,1o 2.
6. The case o a hook shape and a wo- ow shape
In his sec ion we de i e an explici o mula o he K onecke coe icien s in he
case µ= (1e1m2) is a hook and ν= (ν1, ν2) is a wo- ow shape. Gi en a pa i ion
λ, he K onecke coe icien s γλ
µν ell us whe he he poin (e1, ν2) belongs o some
egions in N2de e mined by µ,νand λ.
Using he symme y p ope ies o he K onecke p oduc , we may assume ha
i λ= (1d12d2n3n4) hen n4−n3≤d1. (I n4= 0 hen we should ew i e λas
(1d12d2−12n3). Mo eo e , ou hypo hesis becomes n3−2≤d1.)
Recall ha we deno e he alue o he cha ac e is ic unc ion a p oposi ion P
by (P).
Theo em 16. Le λ,µand νbe pa i ions o n, whe e µ= (1e1m2)is a hook and
ν= (ν1, ν2)is a wo- ow shape. Then he K onecke coe icien s γλ
µν a e gi en by
he ollowing:
1. I λis a one- ow shape, hen γλ
µν =δµ,ν.
2. I λis no con ained in any double hook, hen γλ
µν = 0.
KRONECKER PRODUCT 19
3. Suppose λ= (1d12d2n3n4)is a double hook. Assume ha n4−n3≤d1.
(I n4= 0, hen we should w i e 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. I λis a hook, see Co olla y 14.
P oo . Se X= 1+xand Y= 1+yin he comul iplica ion expansion (2) o ob ain
sλ[(1 −x)(1 + y)] = X
µ,ν
γλ
µνsµ[1 −x]sν[1 + y].(27)
Use (17) and (18) o eplace sµand sνin he igh -hand side o (27), and di ide
by (1 −x) o ob ain
sλ[(1 −x)(1 + y)]
1−x=X
µ=(1e1m2)
ν=(ν1,ν2)
γλ
µν(−x)e1yν21−yν1−ν2+1
1−y.(28)
I λis no con ained in any double hook, hen he poin (3,3) is in λ, and by
Se gee ’s o mula, sλ[(1 −x)(1 + y)] equals ze o.
Since we al eady compu ed he K onecke coe icien s when λis con ained in
a hook, we can assume o he es o his p oo ha λis a double hook. Le
λ= (1d12d2n3n4). (No e: I n4= 0 hen we should w i e λ= (1d12d2−12n4).)
Se u1= 1, u2=y, 1=x, and 2=xy in (19), and mul iply by 1−y
1−xon bo h
sides o he esul ing equa ion.
(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 ha e ha (y−x)(1 −xy)(1 −x) = y−x(1 + y+y2) + x2(1 + y+y2)−x3y.
The e o e, looking a he coe icien o xon bo h sides o he equa ion, we see ha
γλ
µν is ze o i e1is di e en om d1+ 2d2, d1+ 2d2+ 1, d1+ 2d2+ 2,o d1+ 2d2+ 3.
20 MERCEDES H. ROSAS
Le e1=d1+ 2d2o e1=d1+ 2d2+ 3. Since ν2≤n/2, we ha e ha
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 ha e ob ained ha o e1=d1+ 2d2o e1=d1+ 2d2+ 3
γλ
µν = (n3≤ν2−d2≤n4).
Le e1=d1+ 2d2+ 1 o e1=d1+ 2d2+ 2. Since ν2≤ ⌊n
2⌋we ha e ha
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 ha e ob ained ha o e1=d1+ 2d2+ 1 o 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
Co olla y 17. The K onecke co icien s, γλ
µν, whe e µis a hook and νis a wo-
ow shape a e always 0,1,2o 3.
7. Final commen s
The inne p oduc o symme ic unc ions was disco e ed by J. H. Red ield [8]
in 1927, oge he wi h he scala p oduc o symme ic unc ions. He called hem
cup and cap p oduc s, espec i ely. D.E. Li lewood [5, 6] ein en ed he inne
p oduc in 1956.
Mo e ecen ly, I.M. Gessel [3] and A. Lascoux [4] ob ained combina o ial in e -
p e a ions o he K onecke coe icien s in some es ic ed cases; Lascoux in he
case whe e µand νa e hooks, and λa s aigh ableaux, and Gessel in he case
ha µand νa e zigzag shapes and λis an a bi a y skew shape. A. Lascoux in-
e p e ed he K onecke coe icien s, when wo o he shapes a e hooks as coun ing
clases o wo ds unde some equi alence ela ion. We e e o [4] o [2] o a com-
ple e s a emen o his esul s. The Co olla y o Theo em 3 in his pape shows ha
each class o wo ds, unde Lascoux’s equi alence, con ains ei he 0, 1, o 2 di e en
ep esen a i es. I. Gessel wo ked on a mo e gene al amewo k, con empla ing he
occu ence o skew ableaux. I was shown in [2] ha in he case whe e wo o he
pa i ions a e hook shapes, and he hi d one is an a bi a y s aigh shape, his
esul is equi alen o Lascoux’s.
In [2], A.M. Ga sia and J.B. Remmel ounded a way o ela e shu les o pe -
mu a ions and K onecke coe icien s. F om he e hey ob ained a combina o ial
in e p e a ion o he K onecke coe icien s when λis a p oduc o homogeneous
symme ic unc ions, and µand νa e a bi a y skew shapes. They also showed
how Gessel’s and Lascoux’s esul s a e ela ed.
J.B. Remmel [9], [10], and J.B. Remmel and T. Whi ehead [11], ob ained o mu-
las o compu ing he K onecke coe icien s in he same cases conside ed in his
pape . Thei app oach was mainly combina o ial:
Fi s , hey expanded he K onecke p oduc sµ∗sνin e ms o Schu unc ions
using he Ga sia-Remmel algo i hm [2]. The p oblem o compu ing he K onecke
coe icien s was educed o compu ing signed sums o ce ain p oduc s o skew
Schu unc ions.
Then hey ob ained a desc ip ion o he coe icien s ha a ise in he expansion
o he esul ing p oduc o skew Schu unc ions in e ms o coun ing 3-colo ed
diag ams in [9], and [10] o 4-colo ed diag ams in [11]. A his poin , hey educed
he p oblem o compu ing a signed sum o colo ed diag am.
Finally, hey de ined in olu ions on hese signed sums o cancel nega i e e ms,
and ob ained he desi ed o mulas by coun ing classes o es ic ed colo ed diag am.
In gene al, i is no ob ious how o go om he de e mina ion o he K onecke
coe icien s γλ
µ,ν when µand νa e wo- ow shapes ound in his pape , and he one

22 MERCEDES H. ROSAS
ob ained by J.B. Remmel and T. Whi ehead [11]. Bu , in some pa icula cases
his is easy o see. Fo ins ance, when λis also a wo- ow shape, bo h o mulas a e
exac ly he same.
Re e ences
[1] N. Be ge on and A.M. Ga sia, “Se gee ’s Fo mula and he Li lewood-Richa dson Rule,”
Linea and Mul ilinea Algeb a 27, 1990, pp. 79–100.
[2] A.M. Ga sia and J.B. Remmel, “Shu les o pe mu a ions and K onecke p oduc s,” G aphs
Combin. 1985, pp. 217–263.
[3] I.M. Gessel, “Mul ipa i e P-pa i ions and inne p oduc s o Schu unc ions,” Con emp.
Ma h. 1984, pp. 289–302.
[4] A. Lascoux, “P odui de K onecke des ep esen a ions du g oup symme ique,” Lec u e
No es in Ma hema ics 1980, 795, Sp inge Ve lag pp. 319–329.
[5] D.E. Li lewood, “The K onecke p oduc o symme ic g oup ep esen a ions,” J. London
Ma h. Soc. 31, 1956, pp. 89–93.
[6] D.E. Li lewood, “Ple hysm and inne p oduc o S- unc ions,” J. London Ma h. Soc. 32,
1957, pp. 18–22.
[7] I.G. Macdonald, Symme ic Func ions and Hall Polynomials, second edi ion, Ox o d Uni-
e si y P ess, 1995.
[8] J.H. Red ield “The heo y o g oup educed dis ibu ion,” Ame . J. Ma h. 49, 1927, pp.
433-455.
[9] J.B. Remmel, “A o mula o he K onecke p oduc o Schu unc ions o hook shapes,” J.
Algeb a 120, 1989, pp. 100–118.
[10] J.B. Remmel, “Fo mulas o he expansion o he K onecke p oduc s S(m,n)⊗S(1p− , )and
S(1k2l)⊗S(1p− , ),” Disc e e Ma h. 99, 1992, pp. 265–287.
[11] J.B. Remmel and T. Whi ehead, “On he K onecke p oduc o Schu unc ions o wo ow
shapes,” Bull. Belg. Ma h. Soc. Simon S e in 1, 1994, pp. 649–683.
[12] B.E. Sagan, The Symme ic G oup, Wadswo h & B ooks/Cole, Paci ic G o e, Cali o nia,
1991.
Depa men o Ma hema ics, B andeis Uni e si y, Wal ham, MA 02254
E-mail add ess:[email p o ec ed]s.edu