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Skew Littlewood–Richardson rules from Hopf algebras
Thomas Lam, Aaron Lauve, Frank Sottile
To cite this version:
Thomas Lam, Aaron Lauve, Frank Sottile. Skew Littlewood–Richardson rules from Hopf algebras. Billey, Sara and Reiner, Victor. 22nd International Conference on Formal Power
Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AN, 22nd
International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010),
pp.355-366, 2010, DMTCS Proceedings.
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FPSAC 2010, San Francisco, USA
DMTCS proc. AN, 2010, 355–366
Skew Littlewood–Richardson rules
from Hopf algebras
Thomas Lam1†, Aaron Lauve2 , and Frank Sottile2‡
1
Department of Mathematics
University of Michigan
Ann Arbor, MI 48109
2
Department of Mathematics
Texas A&M University
College Station, TX 77843
Abstract. We use Hopf algebras to prove a version of the Littlewood–Richardson rule for skew Schur functions,
which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood–Richardson rules for Schur
P - and Q-functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions,
dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group.
Résumé. Nous utilisons des algèbres de Hopf pour prouver une version de la règle de Littlewood–Richardson pour
les fonctions de Schur gauches, qui implique une conjecture d’Assaf et McNamara. Nous établissons également des
règles de Littlewood–Richardson gauches pour les P - et Q-fonctions de Schur et les fonctions de Schur rubbans non
commutatives, ainsi que des règles de Pieri gauches pour les k-fonctions de Schur, les k-fonctions de Schur duales,
et pour l’homologie de la Grassmannienne affine du groupe symplectique.
Keywords: symmetric functions, Littlewood–Richardson rule, Pieri rule, Hopf algebras, antipode
Assaf and McNamara [AM] recently used combinatorics to give an elegant and surprising formula for
the product of a skew Schur function and a complete homogeneous symmetric function. Their paper
included a conjectural skew version of the Littlewood–Richardson rule, and also an appendix by one of
us (Lam) with a simple algebraic proof of their formula. We show how these formulas and much more
are special cases of a simple formula that holds for any pair of dual Hopf algebras. We first establish
this Hopf-algebraic formula, and then apply it to obtain formulas in some well-known Hopf algebras in
combinatorics.
1
A Hopf algebraic formula
We assume basic familiarity with Hopf algebras, as found in the opening chapters of the book [Mon93].
Let H, H ∗ be a pair of dual Hopf algebras over a field k. This means that there is a nondegenerate pairing
† Supported
‡ Supported
by a Sloan Fellowship, and by NSF grants DMS-0901111 and DMS-0652641.
by the NSF grant DMS-0701050.
c 2010 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France
1365–8050 356
Thomas Lam, Aaron Lauve, and Frank Sottile
h·, ·i : H ⊗ H ∗ → k for which the structure of H ∗ is dual to that of H and vice-versa. For example,
H could be finite-dimensional and H ∗ its linear dual, or H could be graded with each component finitedimensional and H ∗ its graded dual. These algebras naturally act on each other [Mon93, 1.6.5]: suppose
that h ∈ H and a ∈ H ∗ and set
X
X
h * a :=
hh, a2 ia1
and
a * h :=
hh2 , aih1 .
(1)
P
(We use Sweedler notation for the coproduct, ∆h =
h1 ⊗ h2 .) These left actions are the adjoints of
right multiplication: for g, h ∈ H and a, b ∈ H ∗ ,
hg, h * ai = hg · h, ai
and
ha * h, bi = hh, b · ai .
This shows that H ∗ is a left H-module under the action in (1). In fact, H ∗ is a left H–module algebra,
meaning that for a, b ∈ H ∗ and h ∈ H,
X
h * (a · b) =
(h1 * a) · (h2 * b) .
(2)
P
P
Recall that the counit ε : H → k and antipode S : H → H satisfy h1 ·ε(h2 ) = h and h1 ·S(h2 ) =
ε(h) · 1H for all h ∈ H.
Lemma 1 For g, h ∈ H and a ∈ H ∗ , we have
X
(a * g) · h =
(S(h2 ) * a) * (g · h1 ) .
(3)
Proof: Let b ∈ H ∗ . We prove first the formula
X
(h * b) · a =
h1 * (b · (S(h2 ) * a)) .
(4)
(This is essentially (∗) in the proof of Lemma 2.1.4 P
in [Mon93].) Expanding the sum using (2) and
coassociativity, (∆ ⊗ 1) ◦ ∆(h) = (1 ⊗ ∆) ◦ ∆(h) = h1 ⊗ h2 ⊗ h3 , gives
X
X
h1 * (b · (S(h2 ) * a)) =
(h1 * b) · (h2 * (S(h3 ) * a))
X
=
(h1 * b) · ((h2 · S(h3 )) * a)
(5)
= (h * b) · a .
(6)
Here, (5) follows as H ∗ is an H-module and (6) from the antipode and counit conditions.
Note that h(a * g) · h, bi = ha * g, h * bi = hg, (h * b) · ai. Using (4) this becomes
X
X
g,
h1 * (b · (S(h2 ) * a)) =
g · h1 , b · (S(h2 ) * a)
X
=
(S(h2 ) * a) * (g · h1 ), b ,
which proves the lemma, as this holds for all b ∈ H ∗ .
2
Remark 2 This proof is identical to the argument in the appendix to [AM], where h was a complete
homogeneous symmetric function in the Hopf algebra H of symmetric functions.
357
Skew Littlewood–Richardson rules from Hopf algebras
2
Application to distinguished bases
We apply Lemma 1 to produce skew Littlewood–Richardson rules for several Hopf algebras in algebraic
combinatorics. We isolate the common features of those arguments.
In the notation of Section 1, let {Lλ } ⊂ H and {Rλ } ⊂ H ∗ be dual bases indexed by some set P, so
hLλ , Rµ i = δλ,µ for λ, µ ∈ P. Define structure constants for H and H ∗ via
X
X
X
ν
ν
Lλ · Lµ =
bλ,µ
Lν
∆(Lν ) =
cλ,µ
Lλ ⊗ Lµ =
Lν/µ ⊗ Lµ
(7)
Rλ · Rµ =
ν
λ,µ
µ
X
X
X
ν
cλ,µ
Rν
∆(Rν ) =
ν
ν
bλ,µ
Rλ ⊗ Rµ =
Rν/µ ⊗ Rµ .
The skew elements Lν/µ and Rν/µ defined above co-multiply according to
X
X
τ
τ
∆(Lτ /σ ) =
cπ,ρ,σ
Lπ ⊗ Lρ
∆(Rτ /σ ) =
bπ,ρ,σ
Rπ ⊗ Rρ .
π,ρ
(8)
µ
λ,µ
(9)
π,ρ
(Note that the structure of H ∗ can be recovered from the structure of H. Thus, we may suppress the
analogs of (8) and the second formula in (9) in the coming sections.)
Finally, suppose that the antipode acts on H in the L-basis according to the formula
S(Lρ ) = (−1)e(ρ) LρT
(10)
for some functions e : P → N and (·)T : P → P. Then Lemma 1 takes the following form.
Theorem 3 (Algebraic Littlewood–Richardson formula) For any λ, µ, σ, τ ∈ P, we have
X
τ
µ+
Lµ/λ · Lτ /σ =
(−1)e(ρ) cπ,ρ,σ
bλλ−,ρT bµ,π
Lµ+ /λ− .
(11)
π,ρ,λ−,µ+
Swapping L ↔ R and b ↔ c in (11) yields the analog for the skew elements Rµ/λ in H ∗ .
Proof: The actions in (1) together with the second formulas for the coproducts in (7) and (8) show that
Rλ * Lµ = Lµ/λ and Lλ * Rµ = Rµ/λ . Now use (3) and (7)–(10) to obtain
X
τ
Lµ/λ · Lτ /σ = (Rλ * Lµ ) · Lτ /σ =
(−1)e(ρ) cπ,ρ,σ
(LρT * Rλ ) * (Lµ · Lπ )
π,ρ
=
X
+
τ
µ
(−1)e(ρ) cπ,ρ,σ
bµ,π
Rλ/ρT * Lµ+
π,ρ,µ+
=
X
+
τ
µ
(−1)e(ρ) cπ,ρ,σ
bλλ−,ρT bµ,π
(Rλ− * Lµ+ ) .
π,ρ,λ−,µ+
This equals the right hand side of (11), since Rλ− * Lµ+ = Lµ+ /λ− .
2
Remark 4 The condition (10) is highly restrictive. It implies that the antipode S, as a linear map, is
conjugate to a signed permutation matrix. Nevertheless, it holds for the Hopf algebras we consider. More
generally, it holds if either H or H ∗ is commutative, for then S is an involution [Mon93, Cor. 1.5.12].
358
3
Thomas Lam, Aaron Lauve, and Frank Sottile
Skew Littlewood–Richardson rule for Schur functions
The commutative Hopf algebra Λ of symmetric functions is graded and self-dual under the Hall inner
product h·, ·i : Λ ⊗ Λ → Q. A systematic study of Λ from a Hopf algebra perspective appears in [Zel81].
We follow the definitions and notation in Chapter I of [Mac95]. The Schur basis of Λ (indexed by partitions) is self-dual, so (7) and (9) become
X
X
X
ν
ν
sλ · sµ =
cλ,µ
sν
∆(sν ) =
cλ,µ
sλ ⊗ sµ =
sν/µ ⊗ sµ
(12)
ν
µ
λ,µ
∆(sτ /σ ) =
X
τ
cπ,ρ,σ
sπ ⊗ sρ ,
(13)
π,ρ
ν
where the cλ,µ
are the Littlewood–Richardson coefficients and the sν/µ are the skew Schur functions
ν
[Mac95, I.5]. Combinatorial expressions for the cλ,µ
and inner products hsµ/λ , sτ /σ i are derived using
τ
the Hopf algebraic structure of Λ in [Zel81]. The coefficients cπ,ρ,σ
occur in the triple product sπ · sρ · sσ ,
τ
cπ,ρ,σ
= hsπ · sρ · sσ , sτ i = hsπ · sρ , sτ /σ i = hsπ ⊗ sρ , ∆(sτ /σ )i .
Write ρ0 for the conjugate (matrix-transpose) of ρ. Then the action of the antipode is
S(sρ ) = (−1)|ρ| sρ0 ,
(14)
which
is just a twisted form of the fundamental involution ω that sends sρ to sρ0 . Indeed, the formula
P
i
i+j=n (−1) ei hj = δ0,n shows that (14) holds on the generators {hn | n ≥ 1} of Λ. The validity
of (14) follows as both S and ω are algebra maps.
Since cλλ−,ρ0 = 0 unless |ρ| = |λ/λ− |, we may write (11) as
X
−
τ
µ+
(−1)|λ/λ | cπ,ρ,σ
cλλ−,ρ0 cµ,π
sµ+ /λ− .
(15)
sµ/λ · sτ /σ =
π,ρ,λ−,µ+
We next formulate a combinatorial version of (15). Given partitions ρ and σ, form the skew shape ρ ∗ σ
by placing ρ southwest of σ. Thus,
if ρ =
and σ =
then ρ ∗ σ =
.
Similarly, if R is a tableau of shape ρ and S a tableau of shape σ, then R ∗ S is the skew tableau of shape
ρ ∗ σ obtained by placing R southwest of S. Fix a tableau T of shape τ . The Littlewood–Richardson
τ
coefficient cρ,σ
is the number of pairs (R, S) of tableaux of respective shapes ρ and σ with R ∗ S Knuthτ
equivalent to T . See [Ful97, Ch. 5, Cor. 2(v)]. Similarly, cπ,ρ,σ
is the number of triples (P, R, S) of
tableaux of respective shapes π, ρ, and σ with P ∗ R ∗ S Knuth-equivalent to T .
Write sh(S) for the shape of a tableau S and S ≡K T if S is Knuth-equivalent to T .
Lemma 5 Let σ, τ be partitions and fix a tableau T of shape τ . Then
X
∆(sτ /σ ) =
ssh(R− ) ⊗ ssh(R+ ) ,
the sum taken over triples (R−, R+, S) of tableaux with sh(S) = σ and R− ∗ R+ ∗ S ≡K T .
2
359
Skew Littlewood–Richardson rules from Hopf algebras
Note that (µ/λ)0 = µ0 /λ0 and the operation ∗ makes sense for skew tableaux. If S is a tableau of skew
shape µ/λ, put |S| = |µ/λ| = |µ| − |λ|.
Theorem 6 (Skew Littlewood–Richardson rule) Let λ, µ, σ, τ be partitions and fix a tableau T of shape
τ . Then
X
−
sµ/λ · sτ /σ =
(−1)|S | sµ+ /λ− ,
(16)
the sum taken over triples (S −, S +, S) of skew tableaux of respective shapes (λ/λ− )0 , µ+ /µ, and σ such
that S − ∗ S + ∗ S ≡K T .
Remark 7 If T is the unique Yamanouchi tableau of shape τ whose ith row contains only the letter i,
then this is almost Conjecture 6.1 in [AM]. Indeed, in this case S is Yamanouchi of shape σ, so the sum
is really over pairs of tableaux, and this explains the σ-Yamanouchi condition in [AM]. The difference
lies in the tableau S − and the reading word condition in [AM]. It is an exercise in tableaux combinatorics
that there is a bijection between the indices (S −, S + ) of Theorem 6 and the corresponding indices of
Conjecture 6.1 in [AM].
Proof Proof of Theorem 6: We reinterpret (15) in terms of tableaux. Let (R−, R+, S) be a triple of
tableaux of partition shape with sh(S) = σ and R− ∗ R+ ∗ S ≡K T . If sh(R− ) = ρ, then by [Ful97, Ch.
λ0
−
of shape (λ/λ− )0 with S − ≡K R− . Likewise,
5, Cor. 2(i)], cλλ−,ρ0 = c(λ
− )0 ,ρ counts skew tableaux S
+
µ
if sh(R+ ) = π, then cµ,π
counts skew tableaux S + of shape µ+ /µ with S + ≡K R+ . Now (15) may be
written as
X
−
sµ/λ · sτ /σ =
(−1)|S | sµ+ /λ− ,
summing over skew tableaux (R−, R+, S −, S +, S) with R± of partition shape, sh(S) = σ, R− ∗R+ ∗S ≡K
T , sh(S + ) = µ+ /µ, sh(S − ) = (λ/λ− )0 , and S ± ≡K R± .
Finally, note that R± is the unique tableau of partition shape Knuth-equivalent to S ± . Since S − ∗S + ∗S
is Knuth-equivalent to T (by transitivity of ≡K ), we omit the unnecessary tableaux R± from the indices
of summation and reach the statement of the theorem.
2
4
Skew Littlewood–Richardson rule for Schur P - and Q-functions
The self-dual Hopf algebra of symmetric functions has a natural self-dual subalgebra Ω. This has dual
bases the Schur P - and Q-functions [Mac95, III.8], which are indexed by strict partitions λ : λ1 > · · · >
λl > 0. Write `(λ) = l for the length of the partition λ. As in Section 3, the constants and skew functions
in the structure equations
X
X
X
ν
ν
Qλ · Qµ =
gλ,µ
Qν
∆(Qν ) =
fλ,µ
Qλ ⊗ Qµ =
Qν/µ ⊗ Qµ
(17)
ν
λ,µ
∆(Qτ /σ ) =
X
τ
fπ,ρ,σ
Qπ ⊗ Qρ
µ
(18)
π,ρ
have combinatorial interpretations (see below). Also, each basis {Pλ } and {Qλ } is almost self-dual in
ν
ν
that Pλ = 2−`(λ) Qλ and gλ,µ
= 2`(λ)+`(µ)−`(ν) fλ,µ
.
360
Thomas Lam, Aaron Lauve, and Frank Sottile
P
The algebra Ω is generated by the special Q-functions qn = Q(n) := i+j=n hi ej [Mac95, III, (8.1)].
This implies that S(qn ) = (−1)n qn , from which we deduce that
S(Qρ ) = (−1)|ρ| Qρ .
As fλλ−,ρ = 0 unless |ρ| = |λ/λ− |, we may write the algebraic rule (11) as
Qµ/λ · Qτ /σ =
X
(−1)|λ/λ
−
|
+
τ
µ
fπ,ρ,σ
gλλ−,ρ gµ,π
Qµ+ /λ− ,
(19)
π,ρ,λ−,µ+
with a similar identity holding for Pµ/λ · Pτ /σ (swapping P ↔ Q and f ↔ g).
We formulate two combinatorial versions of (19). Strict partitions λ, µ are written as shifted Young
diagrams (where row i begins in column i). Skew shifted shapes λ/µ are defined in the obvious manner:
if λ = 431 =
and µ = 31 =
, then λ/µ =
=
.
In what follows, tableaux means semi-standard (skew) shifted tableaux on a marked alphabet [Mac95,
III.8]. We use shifted versions of the jeu-de-taquin and plactic equivalence from [Sag87] and [Ser],
denoting the corresponding relations by ≡SJ and ≡SP , respectively. Given tableaux R, S, T, we write
R ∗ S ≡SP T when representative words u, v, w (built via “mread” [Ser, §2]) of the corresponding shifted
plactic classes satisfy uv ≡SP w.
Stembridge notes (following [Ste89, Prop. 8.2]) that for a fixed tableau M of shape µ,
ν
fλ,µ
= # skew tableaux L : sh(L) = ν/λ and L ≡SJ M .
(20)
Serrano has a similar description of these coefficients in terms of ≡SP . Fixing
P aτ tableau T of shape τ , it
τ
in Pπ · Pρ · Pσ = τ fπ,ρ,σ
Pτ is given by
follows from [Ser, Cor. 1.15] that the coefficient fπ,ρ,σ
τ
fπ,ρ,σ
= # (P, R, S) : sh(P ) = π, sh(R) = ρ, sh(S) = σ, and P ∗ R ∗ S ≡SP T .
(21)
If T is a tableau of shape λ, write `(T ) for `(λ). The formula relating the g’s and f ’s combines with
(20) and (21) to give our next result.
Theorem 8 (Skew Littlewood–Richardson rule) Let λ, µ, σ, τ be strict partitions and fix a tableau T of
shape τ . Then
X
−
−
+
−
+
Qµ/λ · Qτ /σ =
(−1)|λ/λ | 2`(R )+`(R )+`(λ )+`(µ)−`(λ)−`(µ ) Qµ+ /λ− ,
(22)
the sum taken over quintuples (R−, R+, S −, S +, S) with R± of partition shape, sh(S) = σ, R− ∗ R+ ∗
S ≡SP T , sh(S + ) = µ+ /µ, sh(S − ) = (λ/λ− ), and S ± ≡SJ R± .
2
Serrano conjectures an elegant combinatorial description [Ser, Conj. 2.12 and Cor. 2.13] of the structure
ν
constants gλ,µ
in (17): For any tableau M of shape µ, he conjectured
ν
gλ,µ
= # skew tableaux L : sh(L) = ν/λ and L ≡SP M .
(23)
(Note that if S, T are tableaux, then S ≡SP T does not necessarily imply that S ≡SJ T .) This leads to a
conjectural reformulation of Theorem 8 in the spirit of Theorem 6.
361
Skew Littlewood–Richardson rules from Hopf algebras
Conjecture 1 (Conjectural Skew Littlewood–Richardson rule) Let λ, µ, σ, τ be strict partitions, and
fix a tableau T of shape τ . Then
X
−
Qµ/λ · Qτ /σ =
(−1)|S | Qµ+ /λ− ,
(24)
the sum taken over triples (S −, S +, S) of skew tableaux of respective shapes (λ/λ− ), µ+ /µ, and σ such
that S − ∗ S + ∗ S ≡SP T .
Proof: There is a unique shifted tableau R in any shifted plactic class [Ser, Thm. 2.8]. So the conditions
S ± ≡SP R± and R− ∗ R+ ∗ S ≡SP T in (21) and (23) may be replaced with the single condition
S − ∗ S + ∗ S ≡SP T .
2
5
Skew Littlewood–Richardson rule for noncommutative
Schur functions
The Hopf algebra of noncommutative symmetric functions was introduced, independently, in [GKL+ 95,
MR95] as the (graded) dual to the commutative Hopf algebra of quasisymmetric functions. We consider the dual bases (indexed by compositions) {Fα } of Gessel’s quasisymmetric functions and {Rα } of
noncommutative ribbon Schur functions. The structure constants in
X γ
X γ
X
Rα · Rβ =
bα,β Rγ
∆(Rγ ) =
cα,β Rα ⊗ Rβ =
Rγ/β ⊗ Rβ
(25)
γ
α,β
∆(Rτ /σ ) =
X
β
τ
cπ,ρ,σ
Rπ ⊗ Rρ
(26)
π,ρ,σ
may be given combinatorial meaning via the descent map d : Sn → Γn from permutations to compositions and a section of it w : Γn → Sn . These maps are linked via ribbon diagrams, edge-connected skew
Young diagrams (written in the french style), with no 2 × 2 subdiagram present. By way of example,
→ 3141,
d : 148623795 →
w : 3141 →
→ 789623451.
(In the intermediate step for d(w), new rows in the ribbon begin at descents of w. In the intermediate step
for w(α), the boxes in the ribbon are filled left-to-right, bottom-to-top.)
A ribbon α may be extended by a ribbon β in two ways: affixing β to the rightmost edge or bottommost
edge of α (written α/β and αMβ, respectively):
311/ 31 :
→
,
31M 41 :
→
.
If a ribbon γ is formed from α and β in either of these two ways, we write γ ∈ α♦β. The coefficient
γ
bα,β
is 1, if γ ∈ α♦β, and 0 otherwise. If ∗ is the shifted shuffle product on permutations (see (3.4) in
γ
[MR95]), then the coefficient cα,β
is the number of words w in w(α) ∗ w(β) such that d(w) = γ. The
τ
coefficient cπ,ρ,σ has the analogous description.
362
Thomas Lam, Aaron Lauve, and Frank Sottile
Antipode formulas for the distinquished bases were found, independently, in [Ehr96, MR95]:
S(Fα ) = (−1)|α| Fα 0
and
S(Rα ) = (−1)|α| Rα 0 ,
0
where α 0 is the conjugate of α (in the sense of french style skew partitions). For example, (3141) =
211311. The descriptions of the antipode and structure constants in (25) and (26) give a formula for the
product of two skew ribbon Schur functions.
Theorem 9 (Skew Littlewood–Richardson rule) Let α, β, σ, τ be compositions. Then
X
Rβ/α · Rτ /σ =
(−1)|ρ| Rβ + /α− ,
the sum taken over factorizations α ∈ α− ♦ρ 0 , extensions β + ∈ β ♦π, and words w in the shuffle product
w(π) ∗ w(ρ) ∗ w(σ) such that d(w) = τ .
2
Remark 10 The nonzero skew ribbon Schur functions Rβ/α do not correspond to skew ribbon shapes in
a simple way. For example, 111 is not a (connected) sub-ribbon of 221, yet R221/111 = R2 + R11 6= 0.
Contrast this with the skew functions Fβ/α , where Fβ/α 6= 0 if and only if β ∈ ω ♦α for some ribbon
ω. That is, Fβ/α = Fω . Thus we may view α as the last |α| boxes of the ribbon β and β/α as the
complementary ribbon ω. Interpreting Fβ/α · Fτ /σ alternately as a product of ordinary functions or skew
functions yields the curious identity
X γ
X
β+
τ /σ α
Fβ/α · Fτ /σ =
cβ/α,τ /σ Fγ =
(−1)|ρ| bπ,ρ
cα−,ρ 0 cβ,π
Fβ + /α− .
(27)
γ
6
π,ρ,α−,β +
Skew k-Pieri rule for k-Schur functions
Fix an integer k ≥ 1. Let Λ(k) denote the Hopf subalgebra of the Hopf algebra of symmetric functions
generated by the homogeneous symmetric functions h1 , h2 , . . . , hk . Let Λ(k) denote the Hopf-dual quo(k)
(k)
tient Hopf algebra of symmetric functions. We consider the dual bases {sλ } ⊂ Λ(k) and {Fλ } ⊂ Λ(k)
of k-Schur functions and dual k-Schur functions of [LLMS, LM07], also called strong Schur functions
and weak Schur functions in [LLMS]. The k-Schur functions were first introduced by Lapointe, Lascoux,
and Morse in the context of Macdonald polynomials, and were later shown by Lam to represent Schubert
classes in the affine Grassmannian of SL(k+1, C). We refer the reader to the references in [LLMS].
Here λ varies over all k-bounded partitions, that is, those partitions satisfying λ1 ≤ k. There is an
involution λ 7→ λωk on k-bounded partitions called k-conjugation. We have
(k)
(k)
S(sλ ) = (−1)|λ| sλωk
and
(k)
(k)
S(Fλ ) = (−1)|λ| Fλωk .
(k)
If λ = (r) is a one-part partition, then sλ = hr is a homogeneous symmetric function. We have the
k-Pieri and dual k-Pieri rules [LLMS, LM07] (called weak and strong Pieri rules in [LLMS])
X
X
(k)
(k)
sλ · hr =
s(k)
and
Fλ · hr =
Fµ(k)
(28)
µ
λ
r
µ
λ →r µ
for r ≤ k. Here λ r µ denotes an r-weak strip connecting λ and µ—present if and only if both µ/λ and
µωk /λωk are horizontal r-strips. The notation λ →r µ denotes an r-strong strip as introduced in [LLMS],
363
Skew Littlewood–Richardson rules from Hopf algebras
which we will not define here. The terminology comes from the relationship with the weak and strong
(Bruhat) orders of the affine symmetric group. We remark that there may be distinct strong strips λ →r µ
and (λ →r µ)0 which start and end at the same partition, so that the second Pieri rule of (28) may have
multiplicities. (Strictly speaking, the strong strips in [LLMS] are built on (k + 1)-cores, and our λ →r µ
denotes the strips obtained after applying a bijection between (k + 1)-cores and k-bounded partitions.)
(k)
(k)
We define skew functions sλ/µ and Fλ/µ using (7) and (8). There is an explicit combinatorial descrip(k)
tion of Fλ/µ in terms of the weak tableaux of [LLMS], but only a conjectured combinatorial description
(k)
of sλ/µ [LLMS, Conj. 4.18(3)].
Theorem 11 (Skew k-Pieri (or weak Pieri) rule) For k-bounded partitions λ, µ, and r ≤ k,
X
X
(k)
(k)
sµ/λ · hr =
(−1)j
sµ+ /λ−
i+j=r
µ
(λ )
i
− ωk
µ+
ω
j λ k
τ
, use the formula ∆(hr ) =
Proof: In Theorem 3, take Lτ /σ = hr . For cπ,ρ,σ
bλλ−,ρ ωk
and
µ+
,
bµ,π
P
i+j=r
hi ⊗ hj , and for
2
use (28).
Theorem 12 (Skew dual k-Pieri (or strong Pieri) rule) For k-bounded partitions λ, µ, and r ≤ k,
X
X
(k)
(k)
Fλ/µ · hr =
(−1)j
Fλ+ /µ−
i+j=r
λ →i λ+
(µ− )ωk →j µωk
2
Proof: Identical to the proof of Theorem 11.
As an example, let k = 2, r = 2, µ = (2, 1, 1), and λ = (1). Then Theorem 11 states that
(2)
(2)
(2)
s211/1 · h2 = s2211/1 − s2111 ,
(2)
(2)
(2)
(2)
(2)
(2)
which one can verify using (28) and the expansions s211/1 = s21 + s111 and s2211/1 = 2 s2111 + s221 .
Theorem 12 states that
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
F211/1 · h2 = 3 F222/1 + 5 F2211/1 + 3 F21111/1 + 3 F111111/1 − 2 F221 − 3 F2111 − 2 F11111 .
(2)
(2)
(2)
One can verify that both sides are equal to 6 F221 + 5 F2111 + 4 F11111 .
7
Skew Pieri rule for affine Grassmannian of the symplectic group
Fix n ≥ 1. The Hopf algebra Ω of Section 4 contains a Hopf subalgebra Ω(n) generated by the Schur
P -functions P1 , P3 , . . . , P2n−1 . In [LSS10], it was shown that Ω(n) is isomorphic to the homology ring
H∗ (Gr Sp(2n,C) ) of the affine Grassmannian of the symplectic group Sp(2n, C). A distinguished basis
364
Thomas Lam, Aaron Lauve, and Frank Sottile
(n)
(n)
{Pw } ⊂ Ω(n) , representing the Schubert basis, was studied there. The symmetric functions Pw are
shifted versions of the k-Schur functions of Section 6.
(n)
en0 of affine Grassmannian type C permutations: they
The indexing set for the basis {Pw } is the set C
en . A lower Bruhat order ideal Z ⊂ C
en of the
are the minimal length coset representatives of Cn in C
affine type C Weyl group is defined in [LSS10]. Let Zj ⊂ Z denote those v ∈ Z with length `(v) = j.
For each v ∈ Z, there is a nonnegative integer c(v) ∈ Z≥0 , called the number of components of v. We
note that c(id) = 0. With this notation, for each 1 ≤ j ≤ 2n − 1, we have the Pieri rule [LSS10, Thms.
1.3 and 1.4]
X
(n)
Pw(n) · Pj =
2c(v)−1 Pvw
,
(29)
v∈Zj
en0 , and `(vw) = `(v) + `(w).
where the sum is over all v ∈ Zj such that vw ∈ C
It follows from the discussion in Section 4 that the antipode acts on the Pj by S(Pj ) = (−1)j Pj . We
(n)
define Pw/v using (7).
en0 , and r ≤ 2n − 1,
Theorem 13 (Skew Pieri rule) For w, v ∈ C
X
X
(n)
(n)
2c(u)+c(z)−1 Puw/z−1 v ,
Pw/v · Pr =
(−1)j
i+j=r
u∈Zi
z∈Zj
e 0 , `(uw) = `(u) + `(w), and
where the sum is over all u ∈ Zi and z ∈ Zj such that uw, z −1 v ∈ C
n
−1
`(z v) + `(z) = `(v).
τ
, use the formula
Proof: In Theorem 3, take Lτ /σ = Pr and use (29). For the constants cπ,ρ,σ
∆(Pr ) = 1 ⊗ Pr + Pr ⊗ 1 + 2
X
Pr−j ⊗ Pj .
0<j<r
+
τ
µ
If 0 < j < r, the product cπ,ρ,σ
bλλ−,ρ0 bµ,π
in (11) becomes 2 · 2c(u)−1 · 2c(z)−1 = 2c(u)+c(z)−1 . If j = 0
(resp., j = r), it becomes 1 · 2c(u)−1 · 1 = 2c(u)+c(z)−1 (resp., 1 · 1 · 2c(z)−1 = 2c(u)+c(z)−1 ).
2
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Thomas Lam, Aaron Lauve, and Frank Sottile