Duality and deformations of stable Grothendieck polynomials

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by  Damir Yeliussizov

DUALITY AND DEFORMATIONS OF STABLE GROTHENDIECK
POLYNOMIALS
DAMIRYELIUSSIZOV
6
1
0 Abstract. StableGrothendieckpolynomialscanbeviewedasaK-theoryanalogofSchur
2 polynomials. We extend stable Grothendieck polynomials to a two-parameter version,

p which we call canonical stable Grothendieck functions. These functions have the same
e
structure constants (with scaling) as stable Grothendieck polynomials, and (composing
S
withparameterswitching)areself-dualunderthestandardinvolutiveringautomorphism.

9
We study various properties of these functions, including combinatorial formulas, Schur

] expansions, Jacobi-Trudi typeidentities, and associated Fomin-Greene operators.
O
C
.
h
t
a 1. Introduction
m
[ Stable Grothendieck polynomials are certain symmetric power series first studied by

2 Fomin and Kirillov [5, 6]. These functions arise as a stable limit of Grothendieck polynomi-
v
als introduced by Lascoux and Schu¨tzenberger [14]. The stable Grothendieck polynomial
1
8 Gλ(x1,x2,...) (corresponding to a Grassmannian permutation) can be viewed as a defor-
5
mation and K-theory analog of the Schur function s (x ,x ,...) (while the Grothendieck
1 λ 1 2
0 polynomial is an analog of Schubert polynomial).
.
1 As a symmetric function, G has many similarities with s . For example, it can be
λ λ
0
defined by the following ‘bi-alternant’ formula [9, 11, 18]
6
1
v: det xλij+n−j(1−xi)j−1
i G (x ,...,x ) = 1≤i,j≤n
X λ 1 n h i
(x −x )
1≤i<j≤n i j
r
a
Q
and has a combinatorial presentation given by the generating series
G (x ,x ,...) = (−1)|T|−|λ| x#i’sinT,
λ 1 2 i
T i≥1
X Y
where the sum runs over set-valued tableaux of shape λ; a generalization of semi-standard
Young tableaux (SSYT), where boxes may contain sets of integers [2].
Let Λ be the ring of symmetric functions in infinitely many variables x = (x ,x ,...).
1 2
Denote by Λˆ the completion of Λ which includes infinite linear combinations of the basis
elements (in some distinguished basis of Λ, e.g. Schur functions).
Note that G ∈ Λˆ, for instance G = e −e +e −···, where e is the kth elementary
λ (1) 1 2 3 k
symmetric function. It is remarkable that the product of stable Grothendieck polynomials
1
2 DAMIRYELIUSSIZOV
has a finite decomposition
(1) G G = (−1)|ν|−|λ|−|µ|cν G , cν ∈ Z , |ν| ≥ |λ|+|µ|.
λ µ λµ ν λµ ≥0
ν
X
This result was proved by Buch [2] and he described the coefficients cν combinatorially
λµ
using set-valued tableaux, generalizing the Littlewood-Richardson rule for Schur functions.
Buch studied the Grothendieck ring of the Grassmannian Gr(k,Cn) of k-planes in Cn as
the quotient ring Γ/ G ,λ 6⊆ (n−k)k , where Γ = Z · G is a ring with a basis of
λ λ λ
Grothendieck polynomials ((n−k)k is the partition of rectangular shape k×(n−k)).
(cid:10) (cid:11) L
BythefundamentaldualityisomorphismoftheGrassmannianGr(k,Cn)∼= Gr(n−k,Cn),
structure constants have the symmetry cν = cν′ where λ′ denotes the conjugate of λ.
λµ λ′µ′
Therefore the involutive linear map τ : Λˆ → Λˆ (or Γ → Γ; the completion Γˆ of Γ coincides
with Λˆ) defined on bases by setting τ(Gλ) = Gλ′, is a ring homomorphism. The standard
involutive ringautomorphismω (whichmapse toh )extendedonΛˆ bymappingtheSchur
k k
bases sλ to sλ′, does not lead to self-duality for Grothendieck basis, ω(Gλ)= Jλ 6= Gλ′ [12].
Soanotherfamily{J }hasthesamestructureconstants. Wefirstaskthefollowingquestion.
λ
Question 1.1. Is there a family {G } which has the same structure constants as {G } and
λ λ
is self-dual under the standard involution ω, ω(Gλ) = Gλ′?
e
Our aim is to deform stable Grothendieck polynomials to adjust it to the canonical
e e
involution ω. We will give a concrete construction of these symmetric functions and state
the following basic result.
Theorem 1.2. There is an automorphism φ: Λˆ → Λˆ satisfying ω = φτφ−1.
Hence ωφ(Gλ)= φ(Gλ′) and the symmetric function Gλ := φ(Gλ)∈ Λˆ has the same ring
structure and satisfies the duality ω(Gλ) = Gλ′. As we will see, φ is given explicitly by the
e
substitution x 7→ 2x/(2+x).
e e
There is a comultiplication ∆ :Γ → Γ⊗Γ given by
(2) ∆(G )= (−1)|λ|+|µ|−|ν|dν G ⊗G , dν ∈ Z .
ν λµ λ µ λµ ≥0
λ,µ
X
Both product (1) and coproduct ∆ make Γ a commutative and cocommutative bialgebra
[2]. The completion Γˆ of Γ is a Hopf algebra [12] with the antipode given by S(G (x)) =
λ
ω(G (−x)) [20].
λ
The coproduct of Γ is compatible with the dual family for G via the Hall inner product.
λ
These dual stable Grothendieck polynomials g ∈ Λ were explicitly described by Lam and
λ
Pylyavskyy [12] using reverse plane partitions. We have
g g = (−1)|λ|+|µ|−|ν|dν g .
λ µ λµ ν
ν
X
The constants dν are also symmetric up to diagram transpositions, dν = dν′ and simi-
λµ λµ λ′µ′
larly ω(gλ) 6= gλ′. The dual polynomials {gλ} is a (non-homogeneous) Z-basis of Λ and the
DUALITY AND DEFORMATIONS OF STABLE GROTHENDIECK POLYNOMIALS 3
linear map τ¯ : Λ → Λ given by τ¯(gλ) = gλ′ is a ring homomorphism. Similarly, there is an
automorphism φ¯ : Λ → Λ satisfying ω = φ¯τ¯φ¯−1 and we introduce the polynomials g ∈ Λ
λ
dual to Gλ via the Hall inner product, so they also satisfy the duality ω(gλ) = gλ′. Once
we specify the elements g ,g ,... as (free) generators of (the polynomial ring) Λe, they
(1) (2)
charactereize the automorphism φ¯ (and hence the dual family of the G-funcetions).e
Thefunctions G can beeexetended to G for any permutation w ∈ S , similarly as stable
λ w n
Grothendieck polynomials Gw arise. Here we have the duality ω(Gw) = Gw−1, as well as
e e
ω(G ) = G , where w ∈S is the longest permutation.
w w0ww0 0 n
e e
Moregenerally,thefocusofthispaperisontwo-parameterversionsofstableGrothendieck
e e (α,β) (α,β)
polynomials, the dual symmetric functions G (x ,x ,...), g (x ,x ,...). We call
λ 1 2 λ 1 2
themthecanonical stable Grothendieck functionsandthedual canonical stable Grothendieck
polynomials. In special cases they correspond to:
(0,0) (0,0)
(a) Schur functions s = G = g ; the case α + β = 0 corresponds to certain
λ λ λ
deformed Schur functions;
(0,−1) (0,1)
(b) stable Grothendieck polynomials G = G and its dual g = g ;
λ λ λ λ
(−1,0)
(c) weak stable Grothendieck polynomials J = ω(G ) = G , and its dual j =
λ λ λ′ λ
(1,0)
ω(g )= g ;
λ λ′
(−1/2,−1/2) (1/2,1/2)
(d) and the functions discussed above: G = G , g = g .
λ λ λ λ
The functions G(α,β) ∈ Λˆ, g(α,β) ∈ Λ are non-homogeneous symmetric,
λ λ e e
(α,β) (α,β)
G = s + higher degree terms, g = s + lower degree terms.
λ λ λ λ
(α,β) (−α,−β) (α,β) (−α,−β)
The families {g }, {G } are dual via the Hall inner product hg ,G i =
λ λ λ µ
δ , and the involution acts on them as follows:
λ,µ
(α,β) (β,α) (α,β) (β,α)
ω(G ) = G , ω(g ) = g .
λ λ′ λ λ′
(α,β)
Multiplication of the G is governed by the same structure constants (up to scaling) as
λ
for the polynomials G , we have
λ
G(α,β)G(α,β) = (α+β)|ν|−|λ|−|µ|cν G(α,β)
λ µ λµ ν
ν
X
and the comultiplication ∆ is given by
∆(G(α,β)) = (α+β)|λ|+|µ|−|ν|dν G(α,β) ⊗G(α,β).
ν λµ λ µ
λ,µ
X
There are dual Hopf algebra structures with these dual bases parametrized by α,β and
similar properties.
(α,β)
In some sense, the first new parameter α in G uncovers duality (under the involution
λ
(0,β)
ω) of the β-Grothendieck polynomials G . As we will see, construction of the canon-
λ
(α,β)
ical version G corresponds to an appropriate substitution of variables. For the dual
λ
(α,β)
polynomials g , description appear to be more complicated, especially its combinato-
λ
rial presentation. Combining the ‘unifying’ and duality (conjugation) properties described
4 DAMIRYELIUSSIZOV
above, the reason for calling these symmetric functions as canonical is also the following.
In the specialization (α,β) = (q,q−1), the elements {g(α,β) : ρ is a single row or column}
ρ
(under the involution ω) admit a similar characterization as the Kazhdan-Lusztig canoni-
cal bases. The elements {g(α,β) : k ∈ Z } then characterize the dual functions g(α,β) as
(k) >0 λ
generators for Λ.
(α,β) (α,β)
Let us summarize some further results about the functions G , g .
λ λ
Combinatorial formulas. They are based on several new types of tableaux:
(α,β)
- Hook-valued tableaux for G . Each box of such tableau contains a semistandard
λ
hook, the hooks then ‘weakly increase’ in rows and ‘striclty increase’ in columns
(Section 4). This presentation combines set-valued tableaux [2] (sets are single
column hooks) and weak set-valued (or multiset-valued) tableaux given in [12] for
description of J = ω(G ) (multisets are single row hooks).
λ λ
(α,β)
- Rim border tableaux for g . These tableaux are constructed via a special decom-
λ
position of reverse plane partitions (RPP) into rim hooks on borders of the same
entries (Section 7). Here we also describe equivalent objects, called lattice forests
(α,β)
on RPP. Similarly, for αβ = 0 combinatorics of g corresponds to the known
λ
combinatorial presentations of the dual stable Grothendieck polynomials g and
λ
j = ω(g ) given in [12]. For α = 0, we have generating series running over RPP
λ λ
with a special (column) weight and for β = 0 they run over SSYT with a special
(row) weight.
Combinatorial formulas are accompanied with various Pieri type formulas (Section 8) for
(α,β) (α,β) (α,β) (α,β) (α,β) (α,β)
multiplying G ,g on the functions e , h , G , g , G , g .
λ λ k k (k) (k) (1k) (1k)
(α,β) (β,α)
We prove the duality ω(G ) = G using the method of Fomin and Greene [4] on
λ λ′
noncommutative Schur functions. Our approach is based on Schur operators (Section 5),
which also gives a way to define the functions indexed by skew shapes.
In section 9 we present Schur expansions and related combinatorics. In particular, we
(α,β)
show that g are Schur-positive (i.e. their transition coefficients are polynomials in α,β
λ
with positive integer coefficients). We give combinatorial and determinantal formulas for
connection constants.
Jacobi-Truditypedeterminantalidentities (Section 10). Toobtain them, weuseinforma-
tion given in Schur expansions. Using determinantal formulas for connection constants we
obtain new determinantal identities via the Cauchy-Binet formula. In essence, this method
gives combinatorial proofs of corresponding identities, which we discuss in detail for α= 0.
(α,β) (α,β)
In section 11 we extend G to the functions G indexed by permutations w ∈
λ w
(α,β) (α,β)
S . In section 12 we put the canonical stable Grothendieck polynomials g ,G (and
n λ λ
more generally bases of the ring of symmetric functions) in context of the Kazhdan-Lusztig
theory of canonical bases, we give corresponding characterization and discuss some related
problems.
DUALITY AND DEFORMATIONS OF STABLE GROTHENDIECK POLYNOMIALS 5
Acknowledgements. I am grateful to Pavlo Pylyavskyy for stimulation of this project,
many helpful discussions and insightful suggestions. Initial stages of this work began while
I was visiting the IMA (Institute for Mathematics and its Applications), and the major
part was completed during my visit in the Department of Mathematics at MIT. I am
thankful to Richard Stanley and Alexander Postnikov for their hospitality at MIT and for
helpful conversations. I am grateful to Askar Dzhumadil’daev for his support and helpful
discussions, parts of this work were reported in his seminar. I also thank Sergey Fomin and
Victor Reiner for their comments, and the referees for helpful suggestions.
2. Preliminaries and background on Grothendieck polynomials
2.1. Symmetric functions. We assume some familiarity with basic theory, e.g. [16, 22].
Let Λ be the ring of symmetric functions in infinitely many variables x = (x ,x ,...). The
1 2
elementary symmetric function e and complete homogeneous symmetric function h are
k k
given by
e = x ···x , h = x ···x .
k i1 ik k i1 ik
i1<X···<ik i1≤X···≤ik
The ring Λ is a polynomial ring generated by e ,e ,... (or h ,h ,...) and the standard
1 2 1 2
involutive ring automorphism ω :Λ → Λ maps e to h . 1
k k
Classical bases of Λ are indexed by partitions. A partition is a sequence λ = (λ ≥
1
λ ≥ ...) of nonnegative integers with only finitely many nonzero terms. The weight of
2
a partition λ is the sum |λ| = λ +λ +··· . Any partition λ can be viewed as a Young
1 2
diagram which contains λ boxes in the first row, λ boxes in the second row and so on;
1 2
equivalently it is the set {(i,j) : 1 ≤ i ≤ ℓ,1 ≤ j ≤ λ }, where ℓ = ℓ(λ) is the number of
i
nonzero parts of λ. (We use English notation for Young diagrams.) The partition λ′ with
transposed Young diagram, is the conjugate of λ.
We consider Λ over Z, e.g. with the Schur basis Λ= λZ·sλ. We have ω(sλ) = sλ′ and
Λ is equipped with the (standard) Hall inner product h·,·i :Λ×Λ→ Z which makes Schur
L
functions an orthonormal basis, hs ,s i = δ , where δ is the Kronecker symbol.
λ µ λµ
Denote by Λˆ thecompletion of Λwhich consists of symmetricpower series (of unbounded
degree). For the basis of Schur functions s each element f ∈ Λˆ can uniquely be written as
λ
(possibly an infinite sum) f = a s ,a ∈ Z. The Hall inner product h·,·i and the ring
λ λ λ λ
automorphism ω extend as follows: h·,·i : Λ×Λˆ → Z by hs ,s i = δ ; and ω : Λˆ → Λˆ by
P λ µ λ,µ
ω(sλ) = sλ′.
2.2. Grothendieck polynomials. For a generic parameter β (usually β = ±1), the stable
β
Grothendieck polynomial G (x ,...,x ) is a symmetric polynomial which can be defined as
λ 1 n
1Wewill usually write F orF(x)meaningthat it isF(x ,x ,...). Similarly,F(x/(1−αx)) referstothe
1 2
function F(x /(1−αx ),x /(1−αx ),...). Ifthefunction F(x)isofasingle variablexit will bestated or
1 1 2 2
clear from thecontext.
6 DAMIRYELIUSSIZOV
(see [9, 11, 18])
det xλj+n−j(1−βx )j−1
i i
β 1≤i,j≤n
(3) G (x ,...,x ) = .
λ 1 n h (x −x )i
1≤i<j≤n i j
Combinatorially it is presented as the generaQting power series
(4) Gβ(x ,x ,...) = β|T|−|λ| x#ofi’sinT,
λ 1 2 i
T i≥1
X Y
where the sum runs over set-valued tableaux [2], that are fillings of the boxes of the Young
diagram of λ with nonempty sets of positive integers such that if we replace each set by any
of its elements the resulting tableau is always a semi-standard Young tableau (SSYT), i.e.
the numbers weakly increase from left to right in each row and strictly increase from top
to bottom in each column. In other words, a maximal element in each box of a set-valued
tableau is less or equal than any element in a box to the right (in the same row) and strictly
less than any element in boxes below (in the same column).
β β
Obviously, s = G for β = 0 and we set G = G for β = −1. Note that
λ λ λ λ
β|µ|Gβ = G1(βx ,βx ,...), (−β)|µ|Gβ = G (−βx ,−βx ,...).
λ λ 1 2 λ λ 1 2
Originally, the functions G arose as a stable limit of (more general) Grothendieck poly-
λ
nomials indexed by permutations where the partition λ corresponds to a Grassmannian
permutation [5]. We touch this setting in more detail in section 11 when we consider
(α,β)-deformations of Grothendieck polynomials indexed by permutations.
Buch [2] proved that the product of stable Grothendieck polynomials has a finite decom-
position
G G = (−1)|ν|−|λ|−|µ|cν G , cν ∈ Z
λ µ λµ ν λµ ≥0
ν
X
anddescribedacombinatorialLittlewood-Richardsonruleforcν usingset-valuedtableaux.
λµ
He related the commutative ring Γ = Z·G to the K-theory K◦Gr(k,Cn) of the Grass-
λ λ
mannian Gr(k,Cn) of k-planes in Cn. The Grothendieck group K◦Gr(k,Cn) has a basis
L
[O ] that are classes of the structure sheaves of the Schubert varieties in Gr(k,Cn) indexed
λ
by partitions λ whosediagram fitin therectangle k×(n−k) = (n−k)k. Then, thequotient
ring Γ/hG ,λ 6⊆ (n−k)ki is isomorphic to K◦Gr(k,Cn) via the map G → [O ]; i.e. the
λ λ λ
structure constants in the Grothendieck ring with the basis [O ] are the same:
λ
[O ]·[O ] = (−1)|ν|−|λ|−|µ|cν [O ].
λ µ λµ ν
ν
X
The basis {g } of Λ dual to {G } via the Hall inner product was studied by Lam and
λ λ
Pylyavskyy [12]. They gave its formula as the generating series
#{columnsofT containingi}
(5) g (x ,x ,...) = x ,
λ 1 2 i
T i≥1
XY
DUALITY AND DEFORMATIONS OF STABLE GROTHENDIECK POLYNOMIALS 7
where the sum runs over reverse plane partitions, i.e. entries weakly increase in rows and
columns, of shape λ. This basis of dual stable Grothendieck polynomials agrees with the
coproduct ∆ : Γ → Γ⊗Γ of G and addresses K-homology of Grassmannians.
λ
The functions G ,g are not self-dual under the standard involution ω. This map pro-
λ λ
duces here other symmetric functions J = ω(G ),j = ω(g ). Combinatorially, J is
λ λ λ λ λ
described using weak set-valued tableaux (boxes may now contain multisets) and j using
λ
valued-set tableaux (which are SSYT with a special weighted decomposition) [12].
3. The canonical stable Grothendieck functions G(α,β)
λ
Consider now the ring of symmetric functions Λ and its completion Λˆ over Z[α,β] for
two generic parameters α,β. Let Λˆ be a subring of Λˆ under specialization x = 0 for all
n n+i
i ≥ 1, i.e. with finitely many variables x ,...,x .
1 n
Definition 3.1. Let λ be a partition. Define the canonical stable Grothendieck function
(α,β)
G (x ,...,x ) by the formula
λ 1 n
xλj+n−j(1+βx )j−1
det i i
" (1−αxi)λj #
(α,β) 1≤i,j≤n
(6) G (x ,...,x ) = .
λ 1 n n−j
det x
i
1≤i,j≤n
h i
Here
n−j
det x = (x −x )
i i j
1≤i,j≤n
h i 1≤iY<j≤n
is the Vandermonde determinant.
Since numerator and denominator of the expression above are both skew-symmetric,
G(α,β)(x ,...,x ) ∈ Λˆ is a well-defined symmetric power series in x ,...,x . Note that
λ 1 n n 1 n
G(α,β)(x ,...,x ) = 0, if ℓ(λ) > n. It is easy to see that stability property Λˆ → Λˆ
λ 1 n n+1 n
holds:
(α,β) (α,β)
G (x ,...,x ,0) = G (x ,...,x ).
λ 1 n λ 1 n
Therefore we have an extended symmetric power series G(α,β)(x ,x ,...) ∈ Λˆ in infinitely
λ 1 2
many variables. From definition we obtain that
(α,β)
G = s +higher degree terms,
λ λ
the family {G(α,β)} forms a linearly independent set of elements of Λˆ and every element of
λ
Λˆ can uniquely be written as an infinite linear combination of these functions.
Example 3.2. We can compute (e.g. by induction on n → ∞) that
1+βx
(α,β) i
(7) 1+(α+β)G (x ,x ,...) = ,
(1) 1 2 1−αx
i
i≥1
Y
(8) G(α,β)(x ,x ,...) = αiβjs ,
(1) 1 2 (i|j)
i,j≥0
X
8 DAMIRYELIUSSIZOV
where (i|j) denotes the hook shape partition (i+1,1j).
To obtain both formulas from the example, one can e.g. prove by induction on n → ∞
and using (6) for λ = (1,0,...) that
1+βx
(α,β) (α,β)
1+(α+β)G (x ,...,x ,x) = 1+(α+β)G (x ,...,x ) .
(1) 1 n 1−αx (1) 1 n
(cid:16) (cid:17)
(α,β)
As we see, the function G is a certain deformation of Schur and stable Grothendieck
λ
polynomials. In special cases it corresponds to
(0,0)
• Schur polynomials G = s ;
λ λ
(0,−1) (0,β) β
• stable Grothendieck polynomials G = G , G = G .
λ λ λ λ
Pivotal properties of this function are the following.
(α,β)
Theorem 3.3. The functions G satisfy:
λ
(α,β) (β,α)
(i) self-duality ω(G )= G
λ λ′
(ii) product has the finite decomposition
G(α,β)G(α,β) = (α+β)|ν|−|λ|−|µ|cν G(α,β).
λ µ λ,µ ν
ν
X
The duality (i) will be proved later (in Section 5) using the method of Fomin and Greene
[4] on noncommutative Schur functions. Specializing α+β = −1 in (ii), the multiplicative
(α,β)
structureconstants of G coincide with those of G . This property(ii) willbeclear from
λ λ
Proposition 3.4 below.
(α,β)
3.1. Basic properties of G .
λ
Proposition 3.4. The following formulas hold
x x
(α,β) (0,α+β) 1 2
(9) G (x ,x ,...) = G , ,... ,
λ 1 2 λ 1−αx 1−αx
(cid:18) 1 2 (cid:19)
x x
(α,β) 1 2 (α+β,0)
(10) G , ,... = G (x ,x ,...).
λ 1−βx 1−βx λ 1 2
(cid:18) 1 2 (cid:19)
Proof. It is straightforward to verify these identities for a finite variable functions by the
determinantal formula (6), and then extend it to infinitely many variables. (cid:3)
Note that as a relation between the stable Grothendieck polynomials G and the weak
λ
stable Grothendieck polynomials J , similar formulas were given in [19].
λ
Corollary 3.5. For α+β = 0 we have
x x
(α,−α) 1 2
G (x ,x ,...)= s , ,... ,
λ 1 2 λ 1−αx 1−αx
(cid:18) 1 2 (cid:19)
which multiply by the usual Littlewood-Richardson rule.
DUALITY AND DEFORMATIONS OF STABLE GROTHENDIECK POLYNOMIALS 9
Consider a special case (α,β) → (β/2,β/2). We know that the structure constants cν
λµ
satisfy the symmetry cν = cν′ , and thus the linear map τ : Λˆ → Λˆ given by τ(Gβ) = Gβ
λµ λ′µ′ λ λ′
is a ring homomorphism.
β (β/2,β/2) β β
The function G := G is self-dual under the standard involution, ω(G ) = G .
λ λ λ λ′
Hence combining this with Proposition 3.4, the map φ : Λˆ → Λˆ which sends Gβ 7→ Gβ is
λ λ
an automorphismegiven explicitly by the substitution of variables x 7→ xi aned we heave
i 1−β2xi e
ω = φτφ−1, thus confirming Theorem 1.2 for β = −1.
Remark 3.6. The symmetry of cν also implies that there is another automorphism which
λµ
(α,β) (α,β)
maps G 7→ G , and it coincides with the standard involution ω for α = β.
λ λ′
(α,β) (α,β)
3.2. The elements G ,G . Define the generating series
(k) (1k)
(11) H(α,β)(x;t) := 1+(α+β+t) G(α,β)tk−1,
(k)
k≥1
X
(12) E(α,β)(x;t) := 1+(α+β+t) G(α,β)tk−1.
(1k)
k≥1
X
Proposition 3.7. We have
1+βx 1+(β+t)x
(13) H(α,β)(x;t) = j , E(α,β)(x;t) = j.
1−(α+t)x 1−αx
j j
j≥1 j≥1
Y Y
Proof. It is known that (e.g. [15])
i+k−1
(14) Gβ = βis , Gβ = βi e .
(k) (k−1|i) (1k) i i+k
i≥0 i≥0 (cid:18) (cid:19)
X X
The needed identities can be derived by standard manipulations and the substitutions
β → α+β, x → xj . (cid:3)
j 1−αxj
We now give somevariations on theseseries. Firstnotethat usingequation (7)weobtain
(15) G(α,β)tk−1 = G(α+t,β), G(α,β)tk−1 = G(α,β+t).
(k) (1) (1k) (1)
k≥1 k≥1
X X
Let us define
(α,β) (α,β) (α,β) (α,β) (α,β)
(16) h := 1+(α+β)G , h := G +(α+β)G , k > 0,
0 (1) k (k) (k+1)
(α,β) (α,β) (α,β) (α,β) (α,β)
(17) e := 1+(α+β)G , e := G +(α+β)G , k > 0,
0 (1) k (1k) (1k+1)
or in other words,
(18) H(α,β)(x;t) = h(α,β)tk, E(α,β)(x;t) = e(α,β)tk.
k k
k≥0 k≥0
X X
It is easy to show that
x x
(α,β) (α,β) (α,β) (α,β)
(19) h /h = h , e /e = e .
k 0 k 1−αx k 0 k 1+βx
(cid:18) (cid:19) (cid:18) (cid:19)
10 DAMIRYELIUSSIZOV
We have
E(α,β)(t)H(−β,−α)(−t) = 1,
(α,β) (−β,−α)
which implies the following relation between the elements e ,h :
k ℓ
(20) (−1)ie(α,β)h(−β,−α) = δ .
i n−i n,0
i
X
In particular,
(α,β) (−β,−α) (α,β) (−β,−α)
(21) e h = 1+(α+β)G 1−(α+β)G = 1.
0 0 (1) (1)
(cid:16) (cid:17)(cid:16) (cid:17)
Proposition 3.8. Each of the following four families is algebraically independent:
{G(α,β)|k ∈Z }, {G(α,β)|k ∈ Z }, and for α+β 6= 0, {h(α,β)|k ∈ Z }, {e(α,β)|k ∈ Z }.
(k) >0 (1k) >0 k ≥0 k ≥0
(α,β) (α,β)
Proof. If there is a relation between the elements h (or e ), then by (19) there is a
k k
relation between the elements h (or e ) which is not true. Similarly, if there is a relation
i i
(α,β) (α,β) (α,β)
between G (or G ), then by definitions (16), (17) there is a relation between h
(k) (1k) i
(or e(α,β)). (cid:3)
i
4. Hook-valued tableaux
(α,β)
InthissectionwedescribethefunctionsG combinatoriallyusinghook-valuedtableaux.
λ
A Young diagram is called a hook if it has the form (a|b) = (a+1,1b) for some a,b ≥ 0.
In this case, a is called the arm of the hook and b is called the leg of the hook.
We now present a generalization of SSYT where boxes contain tableaux of hook shapes.
Let max(T) (resp. min(T)) of a tableau T be the maximal (resp. minimal) number
contained in T. For two arbitrary tableaux T ,T define the relations T ≤ T (resp.
1 2 1 2
T < T ) if max(T ) ≤ min(T ) (resp. max(T ) < min(T )). So with these orders we can
1 2 1 2 1 2
say that a sequence of nonempty tableaux is (weakly) increasing.
Definition 4.1. A hook-valued tableau of shape λ is a filling of the Young diagram of λ
with SSYTs (instead of numbers) satisfying the following properties:
(1) each box contains one SSYT of hook shape;
(2) thehooksinsideboxesweakly increasefromlefttorightinrowsandstrictly increase
from top to bottom in columns (with the orders defined above).
An example of such tableau is given in Figure 1. Let a(T) and b(T) be the sums of all
arms and legs, respectively, of hooks in T. The weight of T is then defined as w (α,β) =
T
αa(T)βb(T) and the monomial xT = xai where a is the total number of occurrences of i
i i i
in T. Let HT(λ) be the set of hook-valued tableaux of shape λ.
Q
This setting generalizes (and combines) the notions of set-valued tableaux [2] (set is a
single column hook) and weak set-valued tableaux [12] (multiset is a single row hook), see
Figure 2.