Stability of the Heisenberg Product on Symmetric Functions

0.18 MB·

by  Li Ying

STABILITY OF THE HEISENBERG PRODUCT ON SYMMETRIC
FUNCTIONS
LI YING
7
1 Abstract. The Heisenberg product is an associative product defined on symmetric
0 functions which interpolates between the usual product and the Kronecker product. In
2
1938,MurnaghandiscoveredthattheKroneckerproductoftwoSchurfunctionsstabilizes.

n We prove an analogous result for the Heisenberg product of Schur functions.
a
J

2
1

] 1. Introduction
O
C Aguiar, Ferrer Santos, and Moreira introduced a new product on symmetric functions
. (also on representations of symmetric group) [1]. Unlike the usual product and the Kro-
h
t necker product, the terms appearing in Heisenberg product of two Schur functions have
a
m different degrees. The highest degree component is the usual product. When the Schur
functions have the same degree, the lowest degree component of the Heisenberg product
[

is their Kronecker product. In 1938, Murnaghan [6] found that the Kronecker product of
1
v two Schur functions stabilizes in the following sense. Given a partition λ of l and a large
3 integer n, let λ[n] be the partition of n by prepending a part of size n − l to λ. Given
0
two partitions λ and µ, the coefficients appearing in the Schur expansion of the Kronecker
2
3 product s ∗s do not depend upon n when n is large enough.
λ[n] µ[n]
0
The paper is organized as follows. In the second section, we give some basic notations
.
1 and definitions. In the third section, we define the Aguiar coefficients and prove that
0
low degree terms of the Heisenberg product also stabilize. Section 4 offers an example of
7
1 stabilization. In the last section, we define the stable Aguiar coefficients, and show how
:
v to recover the usual Aguiar coefficients from the stable ones.
i
X
r
a 2. Preliminaries
Webeginbydefining theKronecker coefficients intermsofrepresentations ofsymmetric
groups. For any partition α, let V denote the irreducible representation of S indexed
α |α|
by α. Let λ, µ and ν be partitions of n (written as λ,µ,ν ⊢ n). While the tensor product
V ⊗ V is a representation of S × S , it can also be considered as a representation
λ µ n n
of S (by viewing S as a subgroup of S × S through the diagonal map). Write it
n n n n
as ResSn×Sn(V ⊗ V ). The Kronecker coefficient gν is the multiplicity of V in the
Sn λ µ λ,µ ν
decomposition of ResSn×Sn(V ⊗V ) into irreducibles. That is,
Sn λ µ
ResSSnn×Sn(Vλ ⊗Vµ) = Mgλν,µVν.
ν⊢n
1
2 LI YING
There is one-to-onecorrespondence between irreducible representations and Schur func-
tions by the Frobenius map, which sends V to the Schur function s . So we could also
λ λ
express the Kronecker product (denoted by ∗) in terms of symmetric functions:
sλ ∗sµ = Xgλν,µsν.
ν⊢n
We will switch between the two languages.
Given a partition λ = (λ ,λ ,...) and a positive integer n, let λ[n] be the sequence
1 2
(n−|λ|,λ ,λ ,...). When n ≥ |λ|+λ , λ[n] is a partition of n. The stability of Kronecker
1 2 1
ν[n]
coefficients says that for any partitions λ, µ, ν, the Kronecker coefficient g does
λ[n],µ[n]
not depend on n when n is large enough. Write gν for the stable value of the above
λµ
Kronecker coefficient and call it a reduced Kronecker coefficient. When all the Kronecker
coefficients in s ∗ s reach the reduced ones, we say that the Kronecker product
λ[n] µ[n]
stabilizes. Moreover, Murnahan [6] also claimed that gν vanishes unless
λµ
|λ| ≤ |µ|+|ν|, |µ| ≤ |λ|+|ν|, |ν| ≤ |λ|+|µ|,
which are triangle inequalities. When |ν| = |λ| + |µ|, gν is equal to the Littlewood-
λµ
Richardson coefficient cν [6].
λµ
Briand et al. [2] determined when the expansion of Kronecker product stabilizes and
provide with another condition for the reduced Kronecker coefficient being nonzero.
Proposition 2.1 ([2] Theorem 1.2). Let λ and µ be partitions. The expansion of the
Kronecker product s ∗s stabilizes at n = |λ|+|µ|+λ +µ .
λ[n] µ[n] 1 1
Proposition 2.2 ([2] Theorem 3.2). Let λ and µ be partitions, then
max{|ν|+ν |ν parition, gν > 0} = |λ|+|µ|+λ +µ .
1 λ,µ 1 1
Other than stability, we do not know much about the Kronecker coefficients. Aguiar et
al. [1] introduced a new product which interpolates between the Kronecker and the usual
product on symmetric functions.
Definition 2.1. (Heisenberg product) Let V and W be representations of S and S
n m
respectively. Fix an integer l ∈ [max{m,n},m + n], and let a = l − m, b = n + m− l,
and c = l − n. Observe that S × S is a subgroup of S : S × S ֒→ S , for any
x y x+y x y x+y
nonnegative integers x and y. Also, S can be considered as a subgroup of S ×S through
b b b
the diagonal embedding ∆ : S ֒→ S ×S . We have the diagram of inclusions:
Sb b b b
S ×S ×S ×S (cid:31)(cid:127) // S ×S = S ×S
a b b c a+b b+c n m
(2.1) idSa×∆SSb×i×dScS(cid:31)?OO ×S'(cid:7)✐✐(cid:31)(cid:127)✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐// S44 = S
a b c a+b+c l
The Heisenberg product (denoted by ♯) of V and W is
n+m
(2.2) V♯W = M (V♯W)l,
l=max(n,m)
STABILITY OF THE HEISENBERG PRODUCT ON SYMMETRIC FUNCTIONS 3
where
(2.3) (V♯W) = IndSl ResSn×Sm (V ⊗W)
l Sa×Sb×Sc Sa×Sb×Sc
is the degree l component.
When l = m+ n, (V♯W) = IndSn+m (V ⊗W), which is the usual induction product
l Sn×Sm
of representations (corresponding to the usual product of symmetric functions); when
l = n = m, (V♯W) = ResSl×Sl(V⊗W), whichistheKronecker productofrepresentations.
l Sl
The Heisenberg product connects the usual induction product and the Kronecker product.
Remarkably, this product is associative ([1] Theorem 2.3, Theorem 2.4, Theorem 2.6.) By
the definition of the Heisenberg product, the lower degree components behave like the
Kronecker product. A natural question is whether those components stabilize.
Theorem 2.3. Given nonnegative integers d and h and two partitions λ and µ, the
expansion of (V ♯V ) (degree h higher than the lowest degree) stabilizes when
λ[n] µ[n−d] n+h
n ≥ |λ|+|µ|+λ +µ +3h+2d. Moreover, this is where the stabilization begins.
1 1
3. Proof of Theorem 2.3
Let λ be a partition. Define λ+ to be the partition obtained from λ by adding 1
to the first part λ+ = (λ + 1,λ ,λ ,...); similarly, set λ− = (λ − 1,λ ,λ ,...). Let
1 2 3 1 2 3
λ = (λ ,λ ,...)bethepartitionobtainedfromλbyremoving thefirstpart. Forpartitions
2 3
λ and µ, we set λ+µ = (λ +µ ,λ +µ ,...) and λ−µ = (λ −µ ,λ −µ ,...).
1 1 2 2 1 1 2 2
Lemma 3.1. Let λ, µ and ν be partitions with |ν| = |λ|+|µ|
(1) If ν −ν ≥ |λ|, then cν = cν+ .
1 2 λ,µ λ,µ+
(2) If µ −µ ≥ |λ|, then cν = cν+
1 2 λ,µ λ,µ+
Proof. By the Littlewood-Richardson Rule, cγ (α, β, and γ are partitions) counts the
α,β
number of semistandard skew tableaux of shape γ/β and weight α whose row reading
word is a lattice permutation ([4], Chapter 1 Section 9). Let Tγ be the set of these
α,β
tableaux. We show that |Tν | = |Tν+ |.
λ,µ λ,µ+
Note that Tν = ∅ unless µ ⊂ ν, and µ ⊂ ν if and only if µ+ ⊂ ν+, hence it is enough
λ,µ
to consider the case µ ⊂ ν.
The skew diagrams ν/µ and ν+/µ+ differ only by a shift of the first row. Since ν −ν ≥
1 2
|λ|, the first row (may be empty) of ν/µ is disconnected from the rest of the skew diagram,
and similarly for ν+/µ+. This gives us a natural bijection between Tν and Tν+ . Hence
λ,µ λ,µ+
|Tν | = |Tν+ |, and (1) is proved.
λ,µ λ,µ+
The proof of (2) is the same, as µ −µ ≥ |λ| also implies that the first row of ν/µ is
1 2
(cid:3)
disconnected from the rest of it.
Remark 3.1. When λ, µ, and ν do not satisfy the conditions in Lemma 3.1, the one unit
shift of the first row may fail to be a bijection between Tν and Tν+ . However, it is still
λ,µ λ,µ+
a well defined injection from Tν to Tν+ , which means cν ≤ cν+ . In other words, the
λ,µ λ,µ+ λ,µ λ,µ+
ν[n+|λ|]
sequence c is weakly increasing and is constant when n is large.
λ,µ[n]
4 LI YING
Brion[3] andManivel [5]provide withananalogousresult fortheKronecker coefficients:
Proposition 3.2. Let λ, µ, and ν be partitions. The sequence gν[n] is weakly increas-
λ[n],µ[n]
ing.
The Aguiar coefficient aν is the multiplicity of V in V ♯V , i.e.
λ,µ ν λ µ
n+m
Vλ♯Vµ = M Maνλ,µVν.
l=max(n,m) ν⊢l
and we set aν = 0 if λ, µ or ν is not a partition. The first part of Theorem 2.3 states
λ,µ
that when n ≥ |λ|+|µ|+λ +µ +3h+2d,
1 1
(3.1) aν− = aν
λ[n],µ[n−d] λ[n+1],µ[n−d+1]
for all ν ⊢ n+h+1.
To prove (3.1), we first express the Aguiar coefficient in terms of the Littlewood-
Richardson coefficients and the Kronecker coefficients.
Lemma 3.3. For each ν ⊢ l,
(3.2) aν = X cλ cµ gδ cτ cν
λ,µ α,β η,ρ β,η α,δ τ,ρ
α⊢a,ρ⊢c, τ ⊢n
β,η,δ⊢b
where max(n,m) ≤ l ≤ n+m, a = l−m, b = m+n−l, and c = l −n.
Proof. Consider the diagram (2.1) we used to define the Heisenberg product. Given par-
titions λ ⊢ n and µ ⊢ m, V ⊗ V is a representation of S × S (= S × S ). We
λ µ n m a+b b+c
compute the Heisenberg product of V and V in three steps.
λ µ
(1)
S ×S ×S ×S (cid:31)qq(cid:127) // S ×S = S ×S
a b b c a+b b+c n m
(3.3) S(a2)×(cid:26)(cid:26)S(cid:31)?OOb ×(cid:21)◗Su◗◗c◗%(cid:31)(cid:5)❢(cid:127)◗◗❢(◗❢3◗❢.◗1❢◗)❢◗❢◗❢◗❢(( ❢❢❢❢(❢3❢)❢❢((cid:8)❢❦❢❦❢❦❦❢❦❢(3❦❢.❦❢2❦)❢❦❢❦22❦❦❦// S❦55a+b+c = Sl
S ×S = S ×S
a+b c n c
First, we restrict the representation from S ×S to S ×S ×S ×S ,
n m a b b c
(1) ResSSan××SSbm×Sb×Sc(Vλ ⊗Vµ) = M Mcλα,β cµη,ρVα ⊗Vβ ⊗Vη ⊗Vρ.
α⊢aη⊢b
β⊢bρ⊢c
Second, pull back to S ×S ×S along the diagonal map of S . For α ⊢ a, ρ ⊢ c, and
a b c b
β,η ⊢ b we have,
STABILITY OF THE HEISENBERG PRODUCT ON SYMMETRIC FUNCTIONS 5
(2) ResSSaa××SSbb××SSbc×Sc(Vα ⊗Vβ ⊗Vη ⊗Vρ) = Mgβδ,ηVα ⊗Vδ ⊗Vρ.
δ⊢b
The final step is the induction from S ×S ×S to S (= S ). Break this step into
a b c a+b+c l
two substeps as in (3.3). Given α ⊢ a, δ ⊢ b, and ρ ⊢ c, we have:
(3) IndSl (V ⊗V ⊗V ) = IndSl IndSn×Sc (V ⊗V ⊗V )
Sa×Sb×Sc α δ ρ Sn×Sc Sa×Sb×Sc α δ ρ
= M cτα,δ cντ,ρ Vν.
τ ⊢n
ν ⊢l
Combining (1), (2), and (3) together, gives
(V ⊗V ) = IndSl ResSa×Sb×Sb×ScResSn×Sm (V ⊗V )
λ µ l Sa×Sb×Sc Sa×Sb×Sc Sa×Sb×Sb×Sc λ µ
= M cλα,β cµη,ρ gβδ,η cτα,δ cντ,ρ Vν
α⊢a,ρ⊢c,τ ⊢n
β,η,δ ⊢b,ν ⊢l
So for ν ⊢ l,
aν = X cλ cµ gδ cτ cν ,
λ,µ α,β η,ρ β,η α,δ τ,ρ
α⊢a,ρ⊢c, τ ⊢n
β,η,δ⊢b
(cid:3)
as claimed.
We set cν = 0 when λ, µ, or ν is not a partition. Then (3.2) holds for all compositions
λ,µ
ν of l. Combining (3.1) and (3.2), shows that to prove the first part of Theorem 2.3, it is
enough to show that, when n ≥ |λ|+|µ|+λ +µ +3h+2d,
1 1
X cλ[n] cµ[n−d] gδ cτ cν− =
α,β η,ρ β,η α,δ τ,ρ
(α,β,η,ρ,δ,τ)∈T
(3.4)
X cαλ[∗n,β+∗1] cµη∗[n,ρ+∗1−d] gβδ∗∗,η∗ cτα∗∗,δ∗ cντ∗,ρ∗
(α∗,β∗,η∗,ρ∗,δ∗,τ∗)∈T∗
for all ν ⊢ n+h+1, where
T = {(α,β,η,ρ,δ,τ) | α ⊢ d+h,ρ ⊢ h,τ ⊢ n,β,η,δ ⊢ n−d−h};
T∗ = {(α∗,β∗,η∗,ρ∗,δ∗,τ∗) | α∗ ⊢ d+h,ρ∗ ⊢ h,τ∗ ⊢ n+1,
β∗,η∗,δ∗ ⊢ n−d−h+1}.
Define f : T 7−→ Z and f∗ : T∗ 7−→ Z as follows:
≥0 ≥0
f(α,β,η,ρ,δ,τ) = cλ[n] cµ[n−d] gδ cτ cν−,
α,β η,ρ β,η α,δ τ,ρ
f∗(α∗,β∗,η∗,ρ∗,δ∗,τ∗) = cλ[n+1] cµ[n+1−d] gδ∗ cτ∗ cν .
α∗,β∗ η∗,ρ∗ β∗,η∗ α∗,δ∗ τ∗,ρ∗
6 LI YING
Then equation (3.4) becomes:
X f(α,β,η,ρ,δ,τ) =
(α,β,η,ρ,δ,τ)∈T
(3.5)
X f∗(α∗,β∗,η∗,ρ∗,δ∗,τ∗).
(α∗,β∗,η∗,ρ∗,δ∗,τ∗)∈T∗
Some terms in the sums of (3.5) vanish. Let us consider only the nonvanishing terms.
Let T = Trf−1(0) and T∗ = T∗rf∗−1(0), so that T and T∗ index the nonzero terms.
0 0 0 0
Then (3.5) becomes
X f(α,β,η,ρ,δ,τ) =
(α,β,η,ρ,δ,τ)∈T0
(3.6)
X f∗(α∗,β∗,η∗,ρ∗,δ∗,τ∗).
(α∗,β∗,η∗,ρ∗,δ∗,τ∗)∈T∗
0
Lemma 3.4. The natural embedding ϕ from T to T∗:
ϕ(α,β,η,ρ,δ,τ) = (α,β+,η+,ρ,δ+,τ+)
induces a map ϕ| from T to T∗. Moreover f = f∗ ◦ϕ| .
T0 0 0 T0
T(cid:31)(cid:127) ϕ // T∗
T(cid:31)?OO *(cid:31)(cid:10)(cid:127)♦♦♦♦ϕ♦♦|ϕT♦|0T♦0♦♦♦♦♦// ♦ϕ77 ((cid:31)?TOO )
0 0
Proof. For all (α,β,η,ρ,δ,τ) ∈ T , we show that β, η, δ, and τ have large enough first
0
parts so that we can apply Proposition 2.1 and Lemma 3.1 to the Littlewood-Richardson
coefficients and the Kronecker coefficients appearing in the definition of f.
Since n ≥ |λ|+|µ|+λ +µ +3h+2d, we have
1 1
λ[n] −λ[n] ≥ n−|λ|−λ ≥ |µ|+µ +3h+2d ≥ h+d (= |α|)
1 2 1 1
and
µ[n−d] −µ[n−d] ≥ n−d−|µ|−µ ≥ |λ|+λ +3h+d ≥ h (= |ρ|).
1 2 1 1
Using Lemma 3.1 (1), we get
cλ[n] = cλ[n+1] and cµ[n−d] = cµ[n+1−d].
α,β α,β+ η,ρ η,ρ+
As β ⊂ λ[n], |β| ≤ |λ| < n−d−hand(β) ≤ λ . Similarly, we have |η| ≤ |µ| < n−d−h
1 1
and (η) ≤ µ . Since β and η are both partitions of n − d − h, they can be written as
1 1
β = β[n−d−h] and η = η[n−d−h] respectively. They both have large first parts. More
specifically, we have
n−d−h ≥ |λ|+|µ|+λ +µ +2h+d ≥ |β|+|η|+(β) +(η) .
1 1 1 1
By Proposition 2.1, we have
gδ = gδ+ .
β,η β+,η+
STABILITY OF THE HEISENBERG PRODUCT ON SYMMETRIC FUNCTIONS 7
Followed from proposition 2.2,
|δ|+(δ) ≤ |β|+|η|+(β) +(η)
1 1 1
for otherwise gδ = 0, so thus f. Hence,
β,η
|δ|−δ +δ ≤ |λ|+|µ|+λ +µ
1 2 1 1
which gives us
δ −δ ≥ n−d−h−|λ|−|µ|−λ −µ ≥ 2h+d ≥ |α|
1 2 1 1
Applying Lemma 3.1 (2), we get
cτ = cτ+ .
α,δ α,δ+
Also, by the Littlewood-Richardson rule, we have
τ ≤ δ +|α| and τ ≥ δ .
2 2 1 1
So
τ −τ ≥ δ −(δ +|α|) ≥ 2h+d−h−d = h = |ρ|.
1 2 1 2
Hence, by Lemma 3.1 (2), we get
cν− = cν .
τ,ρ τ+,ρ
So
(3.7) f(α,β,η,ρ,δ,τ) = f∗(ϕ(α,β,η,ρ,δ,τ))(6= 0),
which means ϕ(T ) ⊂ T∗ and f = f∗ ◦ϕ. (cid:3)
0 0
Proof of Theorem 2.3. The map of Lemma 3.4 is reversible as the map (α,β,η,ρ,δ,τ) −→
(α,β−,η−,ρ,δ−,τ−) gives a well-defined injection from T∗ to T . So ϕ| is a bijection
0 0 T0
from T to T∗. With this and (3.7), we prove (3.6), and hence the first part of Theorem
0 0
2.3.
To prove that the lower bound is where the stabilization begins, we just need show that
there is ν ⊢ n+h with ν = ν such that aν 6= 0 when n = |λ|+|µ|+λ +µ +
1 2 λ[n],µ[n−d] 1 1
3h+2d. We use the formula (3.2) for aν 6= 0 (replace λ and ν by λ[n] and µ[n−d]
λ[n],µ[n−d]
respectively, and set l = n+h), and take
α = (a) = (d+h),ρ = (c) = (h),
β = λ[n]−α = (n−|λ|−d−h,λ ,λ ,...) = (|µ|+λ +µ +2h+d,λ ,...),
1 2 1 1 1
η = µ[n−d]−ρ = (n−d−|µ|−h,µ ,µ ,...) = (|λ|+λ +µ +2h+d,µ ,...),
1 2 1 1 1
δ = (β+η)[n−d−h] = (n−d−h−|β|−|η|,β +η ,β +η ,...) = (λ +µ +2h+d,λ +µ ,...),
2 2 3 3 1 1 1 1
τ = (δ ,δ +|α|,δ ,...) = (λ +µ +2h+d,λ +µ +d+h,λ +µ ,...),
1 2 3 1 1 1 1 2 2
ν = (τ ,τ +|ρ|,...) = (λ +µ +2h+d,λ +µ +2h+d,λ +µ ,...).
1 2 1 1 1 1 2 2
By the Pieri Rule, 1 = cλ[n] = cµ[n−d] = cτ = cν , as α and ρ have only one part each.
α,β η,ρ α,δ τ,ρ
Since |δ| = |β|+|η|, we have gδ = gδ = cδ (note that δ = β+η) which is also nonzero
β,η β,η β,η
due to the Littlewood-Richardson Rule.
So aν 6= 0 and ν = ν = λ +λ +2h+d, this proves that n = |λ|+|µ|+λ +
λ[n],µ[n−d] 1 2 1 2 1
µ +3h+2d is where the stabilization begins. (cid:3)
1
8 LI YING
When n < |λ|+|µ|+λ +µ +3h+2d, Lemma 3.4 is not true for some ν. However,
1 1
using Remark 3.1 and Proposition 3.2, we know that the map ϕ in Lemma 3.4 induces an
injection from T to T∗ with f ≤ f∗ ◦ϕ| . This gives us the following corollary:
0 0 T0
Corollary 3.5. Given three partitions λ, ν, and µ and two nonnegative integers d and h,
ν[n+h]
the sequence a is weakly increasing.
λ[n],µ[n−d]
4. Example
We give an example of stabilization of the low degree terms of the Heisenberg product.
Let us take λ = (1,1), µ = (1). We check the stability of the two lowest degree
components of s ♯s :
(1,1)[n] (1)[n−1]
s ♯s = (s +s )+(s +s +2s +s )+(s +s )
1,1,1 1,1 2,1,1,1 1,1,1,1,1 3,1 2,2 2,1,1 1,1,1,1 3 2,1
s ♯s = (s +s +s +s +2s +s +s +s )+
2,1,1 2,1 4,2,1 4,1,1,1 3,3,1 3,2,2 3,2,1,1 3,1,1,1,1 2,2,2,1 2,2,1,1,1
(s +5s +7s +9s +8s +7s +2s )+(s +3s +2s +3s +
5 4,1 3,2 3,1,1 2,2,1 2,1,1,1 1,1,1,1,1 4 3,1 2,2 2,1,1
s )
1,1,1,1
The lowest degree component (s ♯s ) for n ≥ 5:
(1,1)[n] (1)[n−1] n
(s ♯s ) = s +3s +4s +4s +4s +3s +s
3,1,1 3,1 5 5 4,1 3,2 3,1,1 2,2,1 2,1,1,1 1,1,1,1,1
(s ♯s ) = s +3s +4s +4s +2s +5s +3s +s +2s +s
4,1,1 4,1 6 6 5,1 4,2 4,1,1 3,3 3,2,1 3,1,1,1 2,2,2 2,2,1,1 2,1,1,1,1
(s ♯s ) = s +3s +4s +4s +2s +5s +3s +s +s +2s +
5,1,1 5,1 7 7 6,1 5,2 5,1,1 4,3 4,2,1 4,1,1,1 3,3,1 3,2,2 3,2,1,1
s
3,1,1,1,1
(s ♯s ) = s +3s +4s +4s +2s +5s +3s +s +s +2s +
6,1,1 6,1 8 8 7,1 6,2 6,1,1 5,3 5,2,1 5,1,1,1 4,3,1 4,2,2 4,2,1,1
s
4,1,1,1,1
······
We create a table for this:
❍
❍ s
❍❍ ν n n−1 n−2 n−2 n−3 n−3 n−3 n−4 n−4 n−4 n−4
n ❍❍
1 2 1 3 2 1 3 2 2 1
1 1 1 1 2 1 1
1 1 1
1
3 1 1
4 1 3 2 3 1
5 1 3 4 4 2 5 3 1
6 1 3 4 4 2 5 3 1 2 1
7 1 3 4 4 2 5 3 1 1 2 1
8 1 3 4 4 2 5 3 1 1 2 1
Table 1. (s ♯s ) for 3 ≤ n ≤ 8.
(1,1)[n] (1)[n−1] n
The first column gives the values of n. The first row lists all the terms which may
appear in the component, and we use the indexing partitions to denote the corresponding
Schur functions. We color a coefficient red if it reaches the stable value. From Table 1, we
can see that different terms stabilizes at different steps, we give an estimate for this in the
STABILITY OF THE HEISENBERG PRODUCT ON SYMMETRIC FUNCTIONS 9
next section. The stabilization of the lowest degree component happens at n = 7 (using
Theorem 2.3 (d = 1 and h = 0), the stabilization begins at n = 2+1+1+ 1+2 = 7).
When n ≥ 7, we have
(s ♯s ) = s +3s +4s +4s +2s
n−2,1,1 n−2,1 n n n−1,1 n−2,2 n−2,1,1 n−3,3
(4.1) +5s +3s +s +s +2s
n−3,2,1 n−3,1,1,1 n−4,3,1 n−4,2,2 n−4,2,1,1
+s .
n−4,1,1,1,1
The second lowest degree component (s ♯s ) for n ≥ 5:
(1,1)[n] (1)[n−1] n+1
(s ♯s ) = s +7s +13s +16s +7s +24s +16s +7s +13s +
3,1,1 3,1 6 6 5,1 4,2 4,1,1 3,3 3,2,1 3,1,1,1 2,2,2 2,2,1,1
7s +s
2,1,1,1,1 1,1,1,1,1,1
(s ♯s ) = s + 7s + 15s + 17s + 13s + 33s + 19s + 17s +
4,1,1 4,1 7 7 6,1 5,2 5,1,1 4,3 4,2,1 4,1,1,1 3,3,1
16s +26s +10s +8s +7s +2s
3,2,2 3,2,1,1 3,1,1,1,1 2,2,2,1 2,2,1,1,1 2,1,1,1,1,1
(s ♯s ) = s +7s +15s +17s +15s +34s +19s +6s +26s +
5,1,1 5,1 8 8 7,1 6,2 6,1,1 5,3 5,2,1 5,1,1,1 4,4 4,3,1
18s +29s +10s +10s +13s +12s +10s +2s +
4,2,2 4,2,1,1 4,1,1,1,1 3,3,2 3,3,1,1 3,2,2,1 3,2,1,1,1 3,1,1,1,1,1
s +2s +s
2,2,2,2 2,2,2,1,1 2,2,1,1,1,1
(s ♯s ) = s +7s +15s +17s +15s +34s +19s +8s +27s +
6,1,1 6,1 9 9 8,1 7,2 7,1,1 6,3 6,2,1 6,1,1,1 5,4 5,3,1
18s +29s +10s +9s +12s +16s +12s +10s +2s +
5,2,2 5,2,1,1 5,1,1,1,1 4,4,1 4,3,2 4,3,1,1 4,2,2,1 4,2,1,1,1 4,1,1,1,1,1
s +4s +3s +s +2s +s
3,3,3 3,3,2,1 3,3,1,1,1 3,2,2,2 3,2,2,1,1 3,2,1,1,1,1
(s s ) = s +7s +15s +17s +15s +34s +19s +8s +27s +
7,1,1 7,1 10 10 9,1 8,2 8,1,1 7,3 7,2,1 7,1,1,1 6,4 6,3,1
18s +29s +10s +2s +10s +12s +16s +12s +10s +
6,2,2 6,2,1,1 6,1,1,1,1 5,5 5,4,1 5,3,2 5,3,1,1 5,2,2,1 5,2,1,1,1
2s +2s +3s +s +4s +3s +s +2s +s
5,1,1,1,1,1 4,4,2 4,4,1,1 4,3,3 4,3,2,1 4,3,1,1,1 4,2,2,2 4,2,2,1,1 4,2,1,1,1,1
(s ♯s ) = s +7s +15s +17s +15s +34s +19s +8s +27s +
8,1,1 8,1 11 11 10,1 9,2 9,1,1 8,3 8,2,1 8,1,1,1 7,4 7,3,1
18s +29s +10s +2s +10s +12s +16s +12s +10s +
7,2,2 7,2,1,1 7,1,1,1,1 6,5 6,4,1 6,3,2 6,3,1,1 6,2,2,1 6,2,1,1,1
2s +s +2s +3s +s +4s +3s +s +2s +s
6,1,1,1,1,1 5,5,1 5,4,2 5,4,1,1 5,3,3 5,3,2,1 5,3,1,1,1 5,2,2,2 5,2,2,1,1 5,2,1,1,1,1
(s ♯s ) = s +7s +15s +17s +15s +34s +19s +8s +27s +
9,1,1 9,1 12 12 11,1 10,2 10,1,1 9,3 9,2,1 9,1,1,1 8,4 8,3,1
18s +29s +10s +2s +10s +12s +16s +12s +10s +
8,2,2 8,2,1,1 8,1,1,1,1 7,5 7,4,1 7,3,2 7,3,1,1 7,2,2,1 7,2,1,1,1
2s +s +2s +3s +s +4s +3s +s +2s +s
7,1,1,1,1,1 6,5,1 6,4,2 6,4,1,1 6,3,3 6,3,2,1 6,3,1,1,1 6,2,2,2 6,2,2,1,1 6,2,1,1,1,1
······
This computation shows that the second lowest degree component of s ♯s
(1,1)[n] (1)[n−1]
stabilizes at n = 10 (using Theorem 2.3 (d = 1 and h = 1), the stabilization begins at
n = 2+1+1+1+3+2 = 10). When n ≥ 10, we have
(s ♯s ) = s +7s +15s +17s +15s
n−2,1,1 n−2,1 n+1 n+1 n,1 n−1,2 n−1,1,1 n−2,3
+34s +19s +8s +27s +18s
n−2,2,1 n−2,1,1,1 n−3,4 n−3,3,1 n−3,2,2
+29s +10s +2s +10s
n−3,2,1,1 n−3,1,1,1,1 n−4,5 n−4,4,1
(4.2) +12s +16s +12s +10s
n−4,3,2 n−4,3,1,1 n−4,2,2,1 n−4,2,1,1,1
+2s +s +2s +3s +s
n−4,1,1,1,1,1 n−5,5,1 n−5,4,2 n−5,4,1,1 n−5,3,3
+4s +3s +s +2s
n−5,3,2,1 n−5,3,1,1,1 n−5,2,2,2 n−5,2,2,1,1
+s .
n−5,2,1,1,1,1
10 LI YING
5. Stable Aguiar Coefficients
Given partitionsλ, µ, andν, Theorem 2.3 tellsus that thesequence {aν[n+|ν|] }∞
λ[n+|λ|],µ[n+|µ|] n=0
is eventually constant. We write aν for that constant value, and call it a stable Aguiar
λ,µ
coefficient. By the way we define stable Aguiar coefficient, we have
aν = aν[n+|ν|] , for all nonnegative integers n.
λ,µ λ[n+|λ|],µ[n+|µ|]
The reason we restrict n to nonnegative integers is that λ[n+|λ|], µ[n+|µ|] and ν[n+|ν|]
need to be partitions. But we can remove this restriction if we extend the definition of
aν to the case where λ, µ, and ν, starting from the second position, are finite weakly
λ,µ
decreasing sequences of positive integers, i.e. λ ≥ λ ≥ λ ≥ ··· ≥ 0, µ ≥ µ ≥ µ ≥
2 3 4 2 3 4
··· ≥ 0, and ν ≥ ν ≥ ν ≥ ··· ≥ 0. Then we have
2 3 4
aν = aν[n+|ν|] , for all integers n.
λ,µ λ[n+|λ|],µ[n+|µ|]
By the Jacob-Trudi determinant formula:
s = det(h )
λ λj+i−j i,j
where h is the complete homogeneous symmetric function, and we set h = 0 when k
k k
is negative and h = 1. We no longer require λ to be a partition, λ can be any finite
0
integer sequence. Then the Jacob-Trudi determinant will give us 0 or ±1 times some
Schur function.
Murnaghan [6] pointed out that the reduced Kronecker coefficients determine the Kro-
necker product. Analogously, the stable Aguiar coefficients also determine the Heisenberg
product, even for small values of n. Consider the lowest degree component of s ♯s as
2,1,1 2,1
an example. The formula (4.1) gives us (let n = 4)
(s ♯s ) = s +3s +4s +4s +2s +5s
2,1,1 2,1 4 4 3,1 2,2 2,1,1 1,3 1,2,1
(5.1)
+3s +s +s +2s +s .
1,1,1,1 0,3,1 0,2,2 0,2,1,1 0,1,1,1,1
Using Jacob-Trudi determinant, we have
s = −s , s = −s , s = −s , and
1,3 2,2 0,3,1 2,1,1 0,2,1,1 1,1,1,1
s = s = s = 0.
1,2,1 0,2,2 0,1,1,1,1
So (5.1) gives us
(s ♯s ) = s +3s +2s +3s +s ,
2,1,1 2,1 4 4 3,1 2,2 2,1,1 1,1,1,1
which coincides with the result we had in Section 4. This example shows the process to
recover the Aguiar coefficients from the stable ones.
Theorem 5.1. Let λ,µ, and ν be partitions with |ν| ≥ |λ| ≥ |µ|, then
4|ν|−|λ|−|µ|
(5.2) aν = X (−1)i−1aν†i,
λ,µ λ,µ
i=1
where ν†i = (ν −i+1,ν +1,ν +1,...,ν +1,ν ,ν ,...).
i 1 2 i−1 i+1 i+2