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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 al-
gebras. Billey, Sara and Reiner, Victor. 22nd International Conference on Formal Power
Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. Dis-
crete 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‡
1Department of Mathematics
University of Michigan
Ann Arbor, MI 48109
2Department 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.
R´ esum´ e. Nous utilisons des alg` ebres de Hopf pour prouver une version de la r` egle de Littlewood–Richardson pour
les fonctions de Schur gauches, qui implique une conjecture d’Assaf et McNamara. Nous ´ etablissons ´ egalement des
r` egles de Littlewood–Richardson gauches pour les P- et Q-fonctions de Schur et les fonctions de Schur rubbans non
commutatives, ainsi que des r` egles 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 by a Sloan Fellowship, and by NSF grants DMS-0901111 and DMS-0652641.
‡Supported by the NSF grant DMS-0701050.
1365–8050 c 2010 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

356 Thomas Lam, Aaron Lauve, and Frank Sottile
h·,·i:H⊗H∗→kfor which the structure of H∗is dual to that of Hand vice-versa. For example,
Hcould be finite-dimensional and H∗its linear dual, or Hcould be graded with each component finite-
dimensional and H∗its graded dual. These algebras naturally act on each other [Mon93, 1.6.5]: suppose
that h∈Hand a∈H∗and set
h*a := Xhh, a2ia1and a * h := Xhh2, aih1.(1)
(We use Sweedler notation for the coproduct, ∆h=Ph1⊗h2.) These left actions are the adjoints of
right multiplication: for g, h ∈Hand a, b ∈H∗,
hg, h * ai=hg·h, aiand 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,
h * (a·b) = X(h1* a)·(h2* b).(2)
Recall that the counit ε:H→kand antipode S:H→Hsatisfy Ph1·ε(h2) = hand Ph1·S(h2) =
ε(h)·1Hfor all h∈H.
Lemma 1 For g, h ∈Hand a∈H∗, we have
(a*g)·h=X(S(h2)* a)*(g·h1).(3)
Proof: Let b∈H∗. We prove first the formula
(h*b)·a=Xh1*(b·(S(h2)* a)) .(4)
(This is essentially (∗)in the proof of Lemma 2.1.4 in [Mon93].) Expanding the sum using (2) and
coassociativity, (∆ ⊗1) ◦∆(h) = (1 ⊗∆) ◦∆(h) = Ph1⊗h2⊗h3, gives
Xh1*(b·(S(h2)* a)) = X(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
g, Xh1*(b·(S(h2)* a))=X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 hwas a complete
homogeneous symmetric function in the Hopf algebra Hof symmetric functions.

Skew Littlewood–Richardson rules from Hopf algebras 357
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} ⊂ Hand {Rλ} ⊂ H∗be dual bases indexed by some set P, so
hLλ, Rµi=δλ,µ for λ, µ ∈ P. Define structure constants for Hand H∗via
Lλ·Lµ=X
ν
bν
λ,µLν∆(Lν) = X
λ,µ
cν
λ,µLλ⊗Lµ=X
µ
Lν/µ ⊗Lµ(7)
Rλ·Rµ=X
ν
cν
λ,µRν∆(Rν) = X
λ,µ
bν
λ,µRλ⊗Rµ=X
µ
Rν/µ ⊗Rµ.(8)
The skew elements Lν/µ and Rν/µ defined above co-multiply according to
∆(Lτ/σ) = X
π,ρ
cτ
π,ρ,σ Lπ⊗Lρ∆(Rτ/σ) = X
π,ρ
bτ
π,ρ,σ Rπ⊗Rρ.(9)
(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 Hin the L-basis according to the formula
S(Lρ) = (−1)e(ρ)LρT(10)
for some functions e:P → Nand (·)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
Lµ/λ ·Lτ/σ =X
π,ρ,λ− ,µ+
(−1)e(ρ)cτ
π,ρ,σ bλ
λ− ,ρTbµ+
µ,π Lµ+/λ−.(11)
Swapping L↔Rand b↔cin (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
Lµ/λ ·Lτ/σ = (Rλ* Lµ)·Lτ/σ =X
π,ρ
(−1)e(ρ)cτ
π,ρ,σ (LρT* Rλ)*(Lµ·Lπ)
=X
π,ρ,µ+
(−1)e(ρ)cτ
π,ρ,σ bµ+ µ,π Rλ/ρT* Lµ+
=X
π,ρ,λ− ,µ+
(−1)e(ρ)cτ
π,ρ,σ bλ
λ− ,ρTbµ+
µ,π (Rλ−* Lµ+).
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 Hor H∗is commutative, for then Sis an involution [Mon93, Cor. 1.5.12].

358 Thomas Lam, Aaron Lauve, and Frank Sottile
3 Skew Littlewood–Richardson rule for Schur functions
The commutative Hopf algebra Λof symmetric functions is graded and self-dual under the Hall inner
product h·,·i: Λ ⊗Λ→Q. A systematic study of Λfrom 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 parti-
tions) is self-dual, so (7) and (9) become
sλ·sµ=X
ν
cν
λ,µsν∆(sν) = X
λ,µ
cν
λ,µsλ⊗sµ=X
µ
sν/µ ⊗sµ(12)
∆(sτ/σ) = X
π,ρ
cτ
π,ρ,σ sπ⊗sρ,(13)
where the cν
λ,µ are the Littlewood–Richardson coefficients and the sν/µ are the skew Schur functions
[Mac95, I.5]. Combinatorial expressions for the cν
λ,µ and inner products hsµ/λ, sτ /σ iare derived using
the Hopf algebraic structure of Λin [Zel81]. The coefficients cτ
π,ρ,σ occur in the triple product sπ·sρ·sσ,
cτ
π,ρ,σ =hsπ·sρ·sσ, sτi=hsπ·sρ, sτ/σi=hsπ⊗sρ,∆(sτ /σ )i.
Write ρ0for 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 ωthat sends sρto sρ0. Indeed, the formula
Pi+j=n(−1)ieihj=δ0,n shows that (14) holds on the generators {hn|n≥1}of Λ. The validity
of (14) follows as both Sand ωare algebra maps.
Since cλ
λ− ,ρ0= 0 unless |ρ|=|λ/λ−|, we may write (11) as
sµ/λ ·sτ/σ =X
π,ρ,λ− ,µ+
(−1)|λ/λ−|cτ
π,ρ,σ cλ
λ− ,ρ0cµ+
µ,π sµ+/λ−.(15)
We next formulate a combinatorial version of (15). Given partitions ρand σ, form the skew shape ρ∗σ
by placing ρsouthwest of σ. Thus,
if ρ=and σ=then ρ∗σ=.
Similarly, if Ris a tableau of shape ρand Sa tableau of shape σ, then R∗Sis the skew tableau of shape
ρ∗σobtained by placing Rsouthwest of S. Fix a tableau Tof shape τ. The Littlewood–Richardson
coefficient cτ
ρ,σ is the number of pairs (R, S)of tableaux of respective shapes ρand σwith R∗SKnuth-
equivalent to T. See [Ful97, Ch. 5, Cor. 2(v)]. Similarly, cτ
π,ρ,σ is the number of triples (P, R, S)of
tableaux of respective shapes π,ρ, and σwith P∗R∗SKnuth-equivalent to T.
Write sh(S)for the shape of a tableau Sand S≡KTif Sis Knuth-equivalent to T.
Lemma 5 Let σ, τ be partitions and fix a tableau Tof shape τ. Then
∆(sτ/σ) = Xssh(R−)⊗ssh(R+),
the sum taken over triples (R− , R+ , S)of tableaux with sh(S) = σand R−∗R+∗S≡KT.2