Table Of ContentMultiplicities in the Kronecker Product
5
s ∗ s
0
(n−p,p) λ
0
2
l
u
Cristina M. Ballantine∗ Rosa C. Orellana†
J
6 College of the Holy Cross Dartmouth College
2
Worcester, MA 01610 Hanover, NH 03755
]
O [email protected] [email protected]
C
. February 2, 2008
h
t
a
m
[
Abstract
1
v
4 In this paper we give a combinatorial interpretation for the coefficient of sν in the
4 Kronecker product s ∗ s , where λ = (λ ,...,λ ) ⊢ n, if ℓ(λ) ≥ 2p − 1 or
(n−p,p) λ 1 ℓ(λ)
5
λ ≥ 2p−1; that is, if λ is not a partition inside the 2(p−1)×2(p−1) square. For
7 1
0 λ inside the square our combinatorial interpretation provides an upper bound for the
5 coefficients. Ingeneral, weareabletocombinatorially computethesecoefficients forall
0
λwhenn >(2p−2)2. Weusethiscombinatorialinterpretationtogivecharacterizations
/
h
for multiplicity free Kronecker products. We have also obtained some formulas for
t
a special cases.
m
: introduction
v
i Let χλ and χµ be the irreducible characters of S (the symmetric group on n let-
X n
ters) indexed by the partitions λ and µ of n. The Kronecker product χλχµ is defined by
r
a (χλχµ)(w) = χλ(w)χµ(w) for w ∈ S . Then χλχµ is the character that corresponds to the
n
diagonal action of S on the tensor product of the irreducible representations indexed by λ
n
and µ. We have
χλχµ = g χν,
λ,µ,ν
X
ν⊢n
where g is the multiplicity of χν in χλχµ. Hence the coefficients g are non-negative
λ,µ,ν λ,µ,ν
integers.
∗Partially supported by the Fulbright Commission
†Partially supported by the Wilson Foundation
1
By means of the Frobenius map one defines the Kronecker (internal) product on the
Schur symmetric functions by
s ∗s = g s .
λ µ λ,µ,ν ν
X
ν⊢n
A formula for decomposing the Kronecker product is unavailable, although the problem has
been studied for nearly one hundred years. In recent years Lascoux [5], Remmel [7, 8], Rem-
mel and Whitehead [9] and Rosas [11] derived closed formulas for Kronecker products of
Schur functions indexed by two row shapes or hook shapes. Gessel [3] obtained a combi-
natorial interpretation for zigzag partitions. However, a combinatorial interpretation is still
lacking even in the case when both λ and µ are two row partitions.
The objective of this paper is to provide a combinatorial interpretation for the Kronecker
coefficients comparable to the Littlewood-Richardson rule which is defined in terms of the
so-called Littlewood-Richardson tableaux (see section 1 for the definition). In this paper we
give a combinatorial interpretation for the coefficient of s in s ∗s , if λ ≥ 2p−1 or
ν (n−p,p) λ 1
ℓ(λ) ≥ 2p−1, in terms of what we call Kronecker Tableaux. In particular, our combinatorial
interpretation holds for all λ if n > (2p − 2)2. For a general λ, the number of Kronecker
tableaux always gives an upper bound for the Kronecker coefficients. Furthermore, using
our combinatorial rule we obtain that g = 0 whenever the intersection of λ and ν
(n−p,p),λ,ν
has less than p boxes.
The techniques we use to obtain our main theorem are purely combinatorial and rely
both on the Jacobi-Trudi identity and on the Garsia-Remmel rule [4] for decomposing the
Kronecker product of a homogeneous symmetric function and a Schur symmetric function.
One can easily deduce from existing formulas that s ∗s is multiplicity free if and
(n−1,1) λ
only if λ = (ak,bl) or λ = (ak), where a,k,b,l are non-negative integers. If p ≥ 2, we
have used our combinatorial rule to determine the partitions λ for which the Kronecker
product s ∗ s is multiplicity free. We have determined that if n ≥ 6, s ∗ s is
(n−p,p) λ (n−2,2) λ
multiplicityfreeifandonlyifλ = (n),(1n),(n−1,1),(2,1n−2)orλ = (ak),wherea,k arenon-
negative integers. If n ≥ 16, s ∗s is multiplicity free if and only if λ = (n),(1n),(n−
(n−3,3) λ
1,1),(2,1n−2) and if n is even also λ = (n/2,n/2) or λ = (2n/2). If p ≥ 4 and n > (2p−2)2,
then s ∗s is multiplicity free if and only if λ = (n),(1n),(n−1,1),(2,1n−2).
(n−p,p) λ
Other applications of our combinatorial interpretation for g include formulas for
(n−p,p),λ,ν
some special partitions λ and ν. The formulas obtained do not include cancellations and are
easy to program.
The paper is organized as follows. In Section 1 we give preliminary definitions and set the
notationused throughoutthepaper. InSection2we giveavariationoftheRemmel-Whitney
[10] algorithm for expanding the skew Schur function s . We then use this algorithm to
λ/µ
prove that the symmetric function s s − s s , where β = (α − 1,α ,...,α ), is
λ/α α λ/β β 1 2 ℓ(α)
Schur positive if and only if λ ≥ 2α −1. In Section 3 we define the Kronecker tableaux
1 1
and give the combinatorial interpretation for g . In the last section we apply our
(n−p,p),λ,ν
combinatorial rule to give characterizations for multiplicity free s ∗ s . We also give
(n−p,p) λ
2
closed formulas for several special cases. For instance, we give a general formula for the
coefficient of s in the product s ∗s and show that these coefficients are unimodal
(n−t,t) (n−p,p) λ
for some special cases of λ.
Acknowledgement: The authors are grateful to Christine Bessenrodt for useful sugges-
tions.
1 Preliminaries and Notation
Details and proofs for the contents of this section can be found in [12, Chap. 7]. A partition
of a non-negative integer n is a weakly decreasing sequence of non-negative integers, λ :=
(λ ,λ ,··· ,λ ), such that |λ| = λ = n. We write λ ⊢ n to mean λ is a partition ofn. The
1 2 ℓ i
nonzero integers λi are called thePparts of λ. We identify a partition with its Young diagram,
i.e. the array of left-justified squares (boxes) with λ boxes in the first row, λ boxes in the
1 2
second row, and so on. The rows are arranged in matrix form from top to bottom. By the
box in position (i,j) we mean the box in the i-th row and j-th column of λ. The length of
λ, denoted ℓ(λ), is the number of rows in the Young diagram.
Given two partitions λ and µ, we write µ ⊆ λ if and only if ℓ(µ) ≤ ℓ(λ) and µ ≤ λ for
i i
1 ≤ i ≤ ℓ(µ). If µ ⊆ λ, we denote by λ/µ the skew shape obtained by removing the boxes
corresponding to µ from λ. The length and parts of a skew diagram are defined in the same
way as for Young diagrams.
Let D = λ/µ be a skew shape and let a = (a ,a ,··· ,a ) be a sequence of positive
1 2 k
integers such that a = |D| = |λ| − |µ|. A decomposition of D of type a, denoted
i
D1 +···+Dk = D,Pis given by a sequence of shapes
µ = λ(0) ⊆ λ(1)... ⊆ λ(k) = λ,
where D = λ(i)/λ(i−1) and |D | = a .
i i i
A semi-standard Young tableau (SSYT) of shape λ/µ is a filling of the boxes of the skew
shape λ/µ with positive integers so that the numbers weakly increase in each row from left
to right and strictly increase in each column from top to bottom. The type of a SSYT T is
the sequence of non-negative integers (t ,t ,...), where t is the number of i’s in T.
1 2 i
2 2 5 7 8 9
4 5 7 9
2 8
8
9
A SSYT of shape λ = (9,5,2,1,1)/(3,1) and type (0,3,0,1,2,0,2,3,3)
Figure. 1
Given T, a SSYT of shape λ/µ and type (t ,t ,...), we define its weight, denoted w(T),
1 2
to be the monomial obtained by replacing each i in T by x and taking the product over all
i
3
boxes, i.e. w(T) = xt1xt2 ···. The skew Schur function s is defined combinatorially by
1 2 λ/µ
the formal power series
s = w(T),
λ/µ
X
T
where the sum runs over all SSYTs of shape λ/µ. To obtain the usual Schur function one
sets µ = ∅.
For any positive integer n, the Schur function indexed by the partition (n) is called the
n-th homogeneous symmetric function and will be denoted by h . That is,
n
h := s .
n (n)
For any two Young diagrams λ and µ, we let λ × µ denote the diagram obtained by
joining the corners of the leftmost, lowest box in λ, i.e the box in position (ℓ(λ),1), with the
rightmost, highest box of µ, i.e. the box in position (1,µ ).
1
λ×µ where λ = (2,1) and µ = (3,2).
Figure. 2
It follows directly from the combinatorial definition of Schur functions that s = s s .
λ×µ λ µ
One defines similarly A×B, when A and B are skew shapes.
The Littlewood-Richardson coefficients are defined via the Hall inner product on sym-
metric functions (see [12, pg. 306]) as follows:
cλ := hs ,s s i = hs ,s i.
µν λ µ ν λ/µ ν
That is, cλ is the coefficient of s in the product s s . The Littlewood-Richardson rule gives
µν λ µ ν
a combinatorial interpretation for the coefficients cλ . Before we state the rule we recall some
µν
terminology. A lattice permutation is a sequence a a ···a such that in any initial factor
1 2 n
a a ···a , the number of i’s is at least as great as the number of (i+1)’s for all i.
1 2 j
The reverse reading word of a tableau is the sequence of entries of T obtained by reading
the entries from right to left and top to bottom, starting with the first row.
The Littlewood-Richardson rule states that the coefficient cλ is equal to the number of
µν
SSYTs of shape λ/µ and type ν whose reverse reading word is a lattice permutation.
Example: The coefficient of s in s s is 2 since there are two Littlewood-
(5,4,3) (4,3,2) (2,1)
Richardson tableaux of shape (5,4,3)/(2,1) and type (4,3,2):
1 1 1 1 1 1
2 2 2 1 2 2
1 3 3 2 3 3
4
The Kronecker product of Schur functions is defined via the Frobenius characteristic map,
ch, from the center of the group algebra of S to the ring of symmetric functions. For a
n
definition of the Frobenius map see [12, pg. 351]. The map ch is a ring homomorphism and
an isometry. It is known that for any irreducible character χλ of the symmetric group
ch(χλ) = s .
λ
Let χλ and χµ be two irreducible characters of S . The Kronecker product χλχµ is defined
n
for every σ ∈ S by χλχµ(σ) = χλ(σ)χµ(σ). Then
n
χλχµ = g χν.
λ,µ,ν
X
ν⊢n
Hence, using the Frobenius characteristic map, one defines the Kronecker product of Schur
functions by
s ∗s := g s .
λ µ λ,µ,ν ν
X
ν⊢n
Littlewood [6] proved the following identity:
s s ∗s = cη (s ∗s )(s ∗s ),
λ µ η γδ λ γ µ δ
X X
γ⊢|λ|δ⊢|µ|
where cη is the Littlewood-Richardson coefficient, i.e. the coefficient of s in the product
γδ η
s s . Garsia and Remmel [4] generalized this result as follows:
γ δ
(s s )∗s = (s ∗s )(s ∗s ),
A B D A D1 B D2
X
D1+D2=D
|D1|=|A|,|D2|=|B|
where A, B and D are skew shapes and the sum runs over all decompositions of the skew
shape D. By induction we have the following generalization:
(h h ···h )∗s = s ···s ,
n1 n2 nk D D1 Dk
D1+·X··+Dk=D
|Di|=ni
where the sum runs over all decompositions of D of type (n ,n ,...,n ).
1 2 k
The reverse lexicographic filling of µ, rl(µ), is a filling of the Young diagram µ with the
numbers 1,2,...,|µ| so that the numbers are entered in order from right to left and top to
bottom.
5
2 A Schur Positivity Theorem
In this section we consider the Schur positivity of the symmetric function sλ/αsα−sλ/α−sα−,
where α = (α ,...,α ) ⊆ λ with α > α and α− = (α −1,α ,...,α ). More explicitly,
1 ℓ(α) 1 2 1 2 ℓ(α)
we show that this symmetric function is Schur positive if and only if λ ≥ 2α −1. In order
1 1
to prove this result we need a variation of the Remmel-Whitney algorithm for expanding the
skew Schur function s .
λ/µ
Skew Algorithm: The algorithm for computing
s = cλ s
λ/µ µν ν
X
|ν|=|λ|−|µ|
is as follows:
(1) Form the reverse lexicographic filling of µ, rl(µ).
(2) Starting with the Young diagram λ, label |µ| of its outermost boxes with the numbers
|µ|,|µ|−1,...,2,1, starting with |µ|, so that the following conditions are satisfied:
(a) After labelling each box, the unlabelled boxes form a Young diagram.
(b) Suppose that in rl(µ) the box in position (i,j) has label x, where x ≤ |µ|. If
j > 1, let x− be the label in position (i,j −1) in rl(µ). If i < ℓ(µ), let x+ be the
label in position (i + 1,j) in rl(µ). Then in λ, x will be placed to the left and
weakly below (to the SW) of x− and above and weakly to the right (to the NE)
of x+.
(3) Fromeachofthediagramsobtained(with|µ|labelledboxes), remove alllabelledboxes.
The resulting unlabelled diagrams correspond to the summands in the decomposition
of s .
λ/µ
Remark: Suppose (i,j) is the position of x in rl(µ) and (l,m) is the new position of x in
λ. The conditions (a) and (b) impose constraints on l and m. It can be easily verified that
l ≥ i and m ≥ µ −j +1, where µ is the number of boxes in the i-th row of µ.
i i
Example: The decomposition of s , where λ = (4,4,2,2), µ = (3,3):
λ/µ
λ = , rl(µ) = 3 2 1.
6 5 4
First we establish the constraints on the position of each label in λ using the Remark.
6
label position (i,j) in µ position (l,m) in λ position relative to x− and x+
6 (2,1) l ≥ 2 and m ≥ 3−1+1 = 3
5 (2,2) l ≥ 2 and m ≥ 3−2+1 = 2 SW of 6
4 (2,3) l ≥ 2 and m ≥ 3−3+1 = 1 SW of 5
3 (1,1) l ≥ 1 and m ≥ 3−1+1 = 3 NE of 6
2 (1,2) l ≥ 1 and m ≥ 3−2+1 = 2 SW of 3 and NE of 5
1 (1,3) l ≥ 1 and m ≥ 3−3+1 = 1 SW of 2 and NE of 4
6
5 6 6
5
5 6 6
4 4 5
3 3
5 6 6
4 4 5
2 3 3 3
5 6 2 6 6
2
4 4 5 4 5
2 3 3 3
5 6 2 6 6
1 1 1 2
4 4 5 4 5
Thus, s = s +s +s .
λ/µ (2,2,1,1) (3,2,1) (3,3)
The Skew algorithm above follows from the Remmel-Whitney algorithm [10] and the fact
that skewing is the adjoint operation of multiplication, i.e. < s ,s >=< s ,s s >. In
λ/µ ν λ µ ν
some sense we are reversing the steps taken in [10] when expanding the product s s .
µ ν
In order to state our first result we need to recall the definition of lexicographic order on
the set of all partitions. If λ = (λ ,λ ,...,λ ) ⊢ n and µ = (µ ,µ ,...,µ ) ⊢ m, we
1 2 ℓ(λ) 1 2 ℓ(µ)
say that λ is less than µ in lexicographic order, and write λ < µ, if there is a non-negative
l
integer k such that λ = µ for all i = 1,2,...,k and λ < µ . The lexicographic order
i i k+1 k+1
is a total order on the set of all partitions.
7
Lemma 2.1. Consider the partitions λ = (λ ,λ ,...,λ ) and α = (α ,α ,...,α ) such
1 2 ℓ(λ) 1 2 ℓ(α)
that α ⊆ λ. The smallest partition ν in lexicographic order such that s appears in the
ν
expansion of s is the partition obtained by reordering the parts of λ/α in decreasing order,
λ/α
i.e. the parts of ν are λ −α ,λ ,−α ,...,λ −α (α = 0 if i > ℓ(α)) reordered such
1 1 2 2 ℓ(λ) ℓ(λ) i
that ν is a partition. Moreover, the multiplicity of s in the expansion of s is equal to 1.
ν λ/α
Proof. When using the Skew algorithm we obtain the smallest partition in lexicographic
order when the labels in the reverse lexicographic order of α, rl(α), are each placed in the
highest possible row of λ (since we will be removing the largest possible number of boxes
from the top rows of λ). We will show that the partition obtained in this way is precisely
the partition obtained by reordering the rows of λ/α. We argue inductively by the number
of rows of α.
Assume α = (α ). We form the reverse lexicographic order of α. According to the Skew
1
algorithm, the label |α| can be placed in the first row of λ if λ > λ . In general, the highest
1 2
position where we can place |α| is (k,λ ), where k is the positive integer such that λ = λ
k k 1
and λ < λ . We will place the other labels of α to the SW of this position respecting the
k+1 1
rules of the algorithm. We will remove the highest possible horizontal strip (a skew shape
so that no two boxes are in the same column) with α boxes starting with position (k,λ )
1 k
and continuing SW. This also follows from Pieri’s rule [12, Corollary 7.15.9].
Now suppose t is the positive integer such that λ −λ ≥ α and λ −λ < α (i.e.
k k+t 1 k k+t−1 1
label 1 will be placed in the (k+t)-th row). Then the smallest shape in lexicographic order
appearing in the Skew algorithm is
(λ ,...,λ ,λ −(λ −λ ),λ −(λ −λ ),...,λ −(λ −λ ),
1 k−1 k k k+1 k+1 k+1 k+2 k+t−1 k+t−1 k+t
λ −(α −(λ −λ )),λ ,...,λ )
k+t 1 k k+t k+t+1 ℓ(λ)
= (λ ,...,λ ,λ ,λ ,...,λ ,λ −α ,λ ,...,λ ).
1 k−1 k+1 k+2 k+t k 1 k+t+1 ℓ(λ)
Since λ = λ , this is precisely the partition obtained by reordering the rows of λ/(α ) =
k 1 1
(λ −α ,λ ,...,λ ).
1 1 2 ℓ(λ)
Suppose the lemma is true for all partitions with ℓ parts that are contained in λ and let
α = (α ,α ,...,α ,α ) be a partition with ℓ+1 parts such that α ⊆ λ. Thus ℓ(α) = ℓ+1.
1 2 ℓ ℓ+1
When we place the labels from the reverse lexicographic order of α into the boxes of
λ according to the Skew algorithm such that each label is placed in the highest possible
row, we first place the labels of the last row of rl(α). They are placed as in the case
α = (α ) above, starting with placing |α| in position (k,λ ), where k is the positive integer
1 k
such that λ = λ and λ < λ (note that k ≥ ℓ(α)). The remaining labels of the
k ℓ(α) k+1 ℓ(α)
last row of α are placed to the SW of this position forming a horizontal strip with boxes
in the highest possible rows of λ. Observe that this is equivalent to labelling the highest
possible horizontal strip of α boxes in (λ ,λ ,...,λ ) such that the unlabelled
ℓ(α) ℓ(α) ℓ(α)+1 ℓ(λ)
boxes yield a Young diagram, and then adding back the rows λ ,...,λ above the shape
1 ℓ(α)−1
8
(λ ,λ ,...,λ ) (with the labelled boxes). Notice that the unlabelled boxes form the
ℓ(α) ℓ(α)+1 ℓ(λ)
partition which is a rearrangement of (λ ,...,λ −α ,...,λ ). According to the Skew
1 ℓ(α) ℓ(α) ℓ(λ)
algorithm,wenowplacethelabelsinrowℓ(α)−1ofrl(α). Requiringthatthelabelsbeplaced
in the highest rightmost position at each step will automatically satisfy the requirements of
the Skew algorithm. Therefore, in order to obtain the smallest partition in lexicographic
order, we just need to label the highest horizontal strips with α boxes starting at row
ℓ(α)
ℓ(α) and continuing SW, then the highest horizontal strip with α boxes starting at row
ℓ(α)−1
ℓ(α)−1 and continuing SW in the remaining unlabelled boxes, and so on until we label the
highest horizontal strip with α boxes starting at row 1 such that, at each step, removing
1
all labelled boxed yields a Young diagram. See Fig. 3 for an example of placing the labels
of the last 3 rows of rl(α) in λ; the horizontal strip containing a’s is the one corresponding
to row ℓ(α), the strip containing b’s corresponds to row ℓ(α)−1 and the strip containing c’s
corresponds to row ℓ(α)−2.
cc···
bb···
ℓ(α)
cc···
bb···
aa···
cc···
bb···
cc··· aa···
Figure. 3
If we follow this procedure of placing each label as high as possible we obtain the smallest
partition in lexicographic order appearing in the expansion of s . The labels in rows
λ/α
¯
2,3,...,ℓ(α) of rl(α) can only be placed in λ = (λ ,λ ,...,λ ). The unlabelled boxes
2 3 ℓ(λ)
form the partition µ = (λ ,µ ,...,µ ) where (µ ,...,µ ) is the smallest partition in
1 2 ℓ(µ) 2 ℓ(µ)
lexicographic order occurring in s . By induction hypothesis, (µ ,...,µ ) is obtained by
λ¯/α¯ 2 ℓ(µ)
rearranging λ −α ,λ − α ,...,λ − α ,λ ,...λ . If we continue the labelling
2 2 3 3 ℓ(α) ℓ(α) ℓ(α)+1 ℓ(λ)
in λ with the labels in row α of rl(α), the labels are placed in µ = (λ ,µ ,...,µ ). By
1 1 2 ℓ(µ)
the discussion in the previous paragraph, we will be labelling the highest possible horizontal
strip in µ. Hence, removing this strip from µ yields the partition ν which is a rearrangement
of λ − α ,µ ,...,µ . By induction we have that ν is the smallest partition and it is a
1 1 2 ℓ(µ)
rearrangement of λ −α ,λ −α ,...λ −α , where α = 0 for i > ℓ(α).
1 1 2 2 ℓ(λ) ℓ(λ) i
9
Since there is only one way of placing the labels of the reverse lexicographic order of α in
the highest possible rows of λ, the multiplicity of s in s , where ν is the smallest partition
ν λ/α
in lexicographic order, equals 1.
Corollary 2.2. Let α = (α ,α ,...,α ) and β = (β ,β ,...,β ) be arbitrary partitions.
1 2 ℓ(α) 1 2 ℓ(β)
The smallest partition ν in lexicographic order appearing in the expansion of s s is the
α β
partition obtained by concatenating the parts of α and β and reordering them to form a
partition.
Proof. The proof is a direct consequence of the Littlewood-Richardson rule. It also follows
directly from the Remmel-Whitney algorithm for multiplying two Schur functions [10] and
Lemma 2.1.
Definition: A symmetric function is said to be Schur positive (or s-positive) if, when
expanded as a linear combination of Schur functions, all the coefficients are positive.
Consider the product s s . The combinatorial definition of Schur functions implies
λ/α α
that s s is the skew Schur function corresponding to the skew shape µ/η, where
λ/α α
µ = (λ +α ,λ +α ,...λ +α ,λ ,λ ,...,λ ,λ ,...,λ )
1 1 1 2 1 ℓ(α) 1 2 ℓ(α) ℓ(α)+1 ℓ(λ)
and
η = (λ ,...,λ ,α ,α ,...,α ).
1 1 1 2 ℓ(α)
ℓ(α) times
| {z }
This is the skew shape α×λ/α.
Example: Let λ = (6,4,2,2) and α = (3,1). Then s s is the skew Schur function
λ/α α
corresponding to the skew shape α×λ/α below.
Definition: Let α = (α ,α , ...,α ) be any sequence of non-negative integers. A se-
1 2 ℓ(α)
quence a a ···a is an α-lattice permutation if in any initial factor a a ···a , 1 ≤ j ≤ n,
1 2 n 1 2 j
we have for any positive integer i:
the number of i′s+α ≥ the number of (i+1)′s+α .
i i+1
Here α = 0 if i > ℓ(α).
i
Then, if ν = (ν ,ν ,...,ν ) ⊢ n, the multiplicity of s in s s is given by the number
1 2 ℓ(ν) ν λ/α α
of SSYT of shape λ/α and type ν/α := (ν −α ,ν −α ,...,ν −α ,ν ,...,ν )
1 1 2 2 ℓ(α) ℓ(α) ℓ(α)+1 ℓ(ν)
whose reverse reading word is an α-lattice permutation. (If α 6⊆ ν, the multiplicity of s in
ν
s s is 0.)
λ/α α
10
