The notion of a parabolically semistandard tableau is a generalisation of Young tableau, which explains combinatorial aspect of various Howe dualities of type A. We prove a JacobiTrudi type formula for the character of parabolically semistandard tableaux of a given generalised partition shape by using nonintersecting lattice paths.
1. INTRODUCTION
A Schur polynomial is a symmetric polynomial, which plays an important role in algebraic combinatorics and representation theory (we refer the reader to
[6
,
15
,
16]
for general exposition on Schur polynomials). Let
x
_{1}
, . . . ,
x_{n}
be mutually commuting
n
variables. Let
s_{λ}
(
x
_{1}
, . . . ,
x_{n}
) be the Schur polynomial corresponding to a partition
λ
= (
λ
_{1}
, . . . ,
λ_{n}
). It is wellknown that
s_{λ}
(
x
_{1}
, . . . ,
x_{n}
) is the character of a complex irreducible polynomial representation of the general linear group
GL_{n}
(ℂ) whose highest weight corresponds to
λ
. There are several equivalent definitions of
s_{λ}
(
x
_{1}
, . . . ,
x_{n}
). One is the celebrated Weyl character formula, which has been extended to the case of a symmetrizable KacMoody algebra
[9]
. There is a combinatorial formula, where
s_{λ}
(
x
_{1}
, . . . ,
x_{n}
) is given as the weight generating function of the set of Young tableaux of shape
λ
. Another wellknown one is the JacobiTrudi formula
where
h_{k}
(
x
_{1}
, . . . ,
x_{n}
) is the
k
th complete symmetric polynomial. While the above formula is originally due to Jacobi, Gessel and Viennot introduced a new interesting proof in terms of nonintersecting lattice paths
[7]
, which has resulted in various generalizations and applications in combinatorics.
In
[12]
, Kwon introduced a new combinatorial object, which we call parabolically semistandard tableaux, in order to understand the combinatorial aspect of Howe duality of type
A
[8]
. For a generalized partition
λ
of length
n
, the weight generating function
S_{λ}
of parabolically semistandard tableaux of shape
λ
gives the character of an irreducible representation of a general linear Lie (super)algebra 𝔤, which arises from (𝔤,
GL_{n}
(ℂ)duality on various Fock spaces. The character
S_{λ}
includes a usual Schur polynomial as a special case, and it also has a WeylKac type character formula, and a JacobiTrudi type formula (see also
[13]
).
The goal of this paper is to show a JacobiTrudi type formula for
S_{λ}
by using nonintersecting lattice paths. Since a parabolically semistandard tableau is roughly speaking a pair (
S
,
T
) of skewshaped Young tableaux with a common inner shape and each component corresponds to an
n
tuple of nonintersecting lattice paths, the pair (
S
,
T
) corresponds to an
n
tuple of nonintersecting zigzagshaped lattice paths which is obtained by gluing nonintersecting paths associated to
S
and
T
. This is our key observation. Then we apply the arguments similar to
[7]
to obtain a JacobiTrudi type formula for
S_{λ}
.
2. PARABOLICALLY SEMISTANDARD TABLEAUX
2.1. Young tableaux
Let us briefly recall necessary background on Young tableaux (see
[6]
for more details). We denote by ℤ and ℤ
_{>0}
the set of integers and positive integers, respectively. A
partition
is a weakly decreasing sequence of nonnegative integers
λ
= (
λ
_{1}
,
λ
_{2}
, . . .) such that Σ
_{i≥1}
λ_{i}
is finite. We say that
λ
is a partition of
n
if Σ
_{i≥1}
λ_{i}
=
n
and denote by
ℓ
(
λ
) the number of positive entries of
λ
. Let
be the set of all partitions, and put
for
n
≥ 1.
A Young diagram
is a collection of boxes arranged in leftjustified row, with weakly decreasing number of boxes in each row from top to bottom. A Young diagram determines a unique partition
λ
= (
λ
_{1}
,
λ
_{2}
, · · ·), where
λ_{i}
is the number of boxes in the
i
th row of the diagram. From now on, we identify a Young diagram with its partition.
Example 2.1.
The Young diagram corresponding to the partition
λ
= (5, 3, 3, 1) is
Let
be given. A
Young tableau T
is a filling of
λ
or the boxes in its Young diagram with positive integers such that the entries are weakly increasing from left to right in each row, and strictly increasing from top to bottom in each column. We say that
λ
is the shape of
T
, and write sh(
T
) =
λ
.
Example 2.2.
For
λ
= (5, 3, 3, 1)
is a Young tableau of shape
λ
.
For
with
λ
⊃
µ
(that is,
λ_{i}
≥
µ_{i}
for all
i
),
λ
/
µ
denotes the
skew Young diagram
. A
skew Young tableau
is a filling of a skew Young diagram
λ
/
µ
with positive integers in the same way as in the case of Young tableaux.
Example 2.3.
For
λ
/
µ
= (5, 3, 3, 1)/(2, 1),
is a skew Young tableau of shape
λ
/
µ
.
Let x = {
x
_{1}
,
x
_{2}
, . . .} be a set of formal commuting variables. For a Young tableau
T
, we put
, where
m_{i}
is the number of times
i
occurs in
T
. For
T
in Example 2.2, we have
. Let
s_{λ}
(x) = Σ
_{T}
x
^{T}
be the
Schur function
corresponding to
, where the sum is over all Young tableaux
T
of
sh
(
T
) =
λ
. For
k
≥ 0, let
h_{k}
(
x
) = s
_{(k)}
(x), which is called the
k
th
complete symmetric function
. For
, we put
h_{µ}
(x) =
h
_{µ1}
(x)
h
_{µ2}
(x). . ..
There is another wellknown equivalent definition of a Schur function called the
JacobiTrudi formula
, which expresses a Schur function as a determinant, and hence as a linear combination of
h_{µ}
(x)’s for
(cf.
[6]
).
Theorem 2.4.
For
with
ℓ
(
λ
) ≤
n
,
where we assume that
h
_{−k}
(x) = 0
for k
≥ 1.
2.2. Parabolically semistandard tableaux
Let 𝓐 be a linearly ordered countable set with a ℤ
_{2}
grading 𝓐 = 𝓐
_{0}
⨆ 𝓐
_{1}
. For
a
∈ 𝓐,
a
is called
even
(resp.
odd
) if
a
∈ 𝓐
_{0}
(resp.
a
∈ 𝓐
_{1}
). Let
λ
/
µ
be a skew Young diagram. A tableau
T
obtained by filling
λ
/
µ
with entries in 𝓐 is called 𝓐
semistandard
if the entries in each row (resp. column) are weakly increasing from left to right (resp. from top to bottom), and the entries in 𝓐
_{0}
(resp. 𝓐
_{1}
) are strictly increasing in each column (resp. row). We say that
λ
/
µ
is the
shape of T
, and write sh(
T
) =
λ
/
µ
. We denote by
SST
_{𝓐}
(
λ
/
µ
) the set of all 𝓐semistandard tableaux of shape
λ
/
µ
. We set
. Let x
_{𝓐}
= {
x_{a}

a
∈ 𝓐 } be a set of formal commuting variables indexed by 𝓐. For
T
∈
SST_{𝓐}
(
λ
/
µ
), put
, where
m_{a}
is the number of occurrences of
a
in
T
. We define the character of
SST
_{𝓐}
(
λ
/
µ
) to be
.
We assume that ℤ
_{>0}
is given with a usual linear ordering and all entries even. When 𝓐 = ℤ
_{>0}
, an 𝓐semistandard tableau is a (skew) Young tableau, and
s_{λ}
(x
_{𝓐}
) is the Schur function associated to
.
Let
be the set of all
generalized partitions
of length
n
. We may identify
λ
with a
generalized Young diagram
as in the following example.
Example 2.5.
The generalized partition
corresponds to
Suppose that 𝓐 and 𝓑 are two disjoint linearly ordered ℤ
_{2}
graded countable sets. Now, let us introduce our main combinatorial object.
Definition 2.6
(
[12]
). For
, a
parabolically semistandard tableau of shape λ
(with respect to (𝓐, 𝓑)) is a pair of tableaux (
T
^{+}
,
T
^{−}
) such that
for some integer
d
≥ 0 and
satisfying (1)
, and (2)
µ
⊂ (
d^{n}
),
µ
⊂
λ
+(
d^{n}
). We denote by
SST_{𝓐/𝓑}
(
λ
) the set of all parabolically semistandard tableaux of shape
λ
with respect to (𝓐, 𝓑).
Roughly speaking, a parabolically semistandard tableau of shape
λ
is a pair of 𝓐semistandard tableau and 𝓑semistandard tableau whose shapes are not necessarily fixed ones but satisfy certain conditions determined by
λ
.
Example 2.7.
Suppose that 𝓐 = ℤ
_{>0}
= { 1 < 2 < 3 < . . . } and 𝓑 = ℤ
_{<0}
= { −1 < −2 < −3 < . . . } with all entries even. Then
where the vertical lines in
T
^{+}
and
T
^{−}
correspond to the one in the generalized partition (3, 2, 0, −2). In this case, we have sh(
T
^{+}
) = ((3, 2, 0, −2) + (3
^{4}
))/(2, 1, 0, 0), and sh(
T
^{−}
) = (3
^{4}
) / (2, 1, 0, 0).
For
, we define the
character of SST_{𝓐/𝓑}
(
λ
) to be
We put
. For
k
∈ ℤ, we put
.
2.3. Irreducible characters
Let us briefly recall a representation theoretic meaning of parabolically semistandard tableaux. For an arbitrary ℤ
_{2}
graded linearly ordered set 𝓒, we denote by
V
_{𝓒}
a superspace with basis {
v_{c}

c
∈ 𝓒 }, and let 𝔤𝔩(
V
_{𝓒}
) be the general linear Lie superalgebra spanned by
E_{cc'}
for
c
,
c'
∈ 𝓒. Here
E_{cc'}
is the matrix where the entry at (
c
,
c'
)position is 1 and 0 elsewhere.
Let 𝔤 = 𝔤𝔩(
V
_{𝓒}
) with 𝓒 = 𝓑 ∗ 𝓐, where 𝓑 ∗ 𝓐 is the ℤ
_{2}
graded set 𝓐 ⨆ 𝓑 with the extended linear ordering defined by
y
<
x
for all
x
∈ 𝓐 and
y
∈ 𝓑. Let
be the super symmetric algebra generated by
, where
is the restricted dual space of
V
_{𝓑}
. One can define a semisimple action of 𝔤 on
, and a semisimple action of
GL_{n}
(ℂ) on
for
n
≥ 1 so that we have the following multiplicityfree decomposition as a (𝔤,
GL_{n}
(ℂ))module,
for a subset
H_{n}
of
, where
L_{n}
(
λ
) is the irreducible
GL_{n}
(ℂ)module with highest weight
λ
∈
H_{n}
, and
L
(
λ
) is an irreducible 𝔤module corresponding to
L_{n}
(
λ
) (see the arguments in [4, Sections 5.1 and 5.4]). We define the character ch
L
(
λ
) to be the trace of the operator
on
L
(
λ
) for
λ
∈
H_{n}
. Finally from a Cauchy type identity for parabolically semistandard tableaux [12, Theorem 4.1], we can conclude the following (cf. [14, Theorem 2.3]).
Theorem 2.8.
For n
≥ 1,
we have
as a (𝔤,
GL_{n}
(ℂ))
module
,
that is
,
,
and the irreducible character
ch
L
(
λ
)
is given by
for
.
Recall that when 𝓐 is finite with 𝓐 = 𝓐
_{0}
or 𝓐
_{1}
and
, the decomposition in Theorem 2.8 is the classical (
GL_{ℓ}
(ℂ),
GL_{n}
(ℂ))Howe duality on the symmetric algebra or exterior algebra generated by ℂ
^{ℓ}
⊗ ℂ
^{n}
, where
ℓ
= 𝓐 (cf.
[8]
). Moreover, the decomposition in Theorem 2.8 includes other Howe dualities of type
A
which have been studied in
[1
,
2
,
3
,
5
,
8
,
10
,
11]
under suitable choices of 𝓐 and 𝓑 (see
[12]
for more details).
3. JACOBITRUDI FORMULA
 3.1. Lattice paths
Definition 3.1.
A
lattice path
is a sequence
of points
v
_{1}
, ...,
v_{r}
in ℤ × ℤ with
v_{i}
= (
a_{i}
,
b_{i}
) such that
b
_{1}
< 0 <
b_{r}
, and
Example 3.2.
The following path
is the lattice path
We denote by 𝒫 the set of lattice paths. Let
p
=
v
_{1}
...
v_{r}
∈ 𝒫 be given with
v_{i}
= (
a_{i}
,
b_{i}
) for 1 ≤
i
≤
r
. We often identify
p
with its extended lattice path
v
_{0}
v
_{1}
...
v_{r}v
_{r+1}
, where
v
_{0}
= (
a
_{1}
, −∞) and
v
_{r+1}
= (
a_{r}
, ∞). Here we regard (
a
_{1}
, −∞) as a point below (
a
_{1}
,
y
) for all
y
≤
b
_{1}
, and (
a_{r}
, ∞) as a point above (
a_{r}
,
y
) for all
y
≥
b^{r}
. We also write
p
:
v
_{0}
→
v
_{r+1}
. For 0 ≤
i
≤
r
, let
v_{i}v
_{i+1}
denote the line segment joining
v_{i}
and
v
_{i+1}
, where we understand
v
_{0}
v
_{1}
(resp.
v_{r}v
_{r+1}
) as an halfinfinite line joining (
a
_{1}
,
b
_{1}
) and (
a
_{1}
, −∞) (resp. (
a_{r}
,
b_{r}
) and (
a_{r}
, ∞)). Let
z
= {
z_{i}

i
∈ ℤ
^{×}
} be a set of formal commuting variables, where ℤ
^{×}
= ℤ \ {0}. We consider a weight monomial
Example 3.3.
For a lattice path
its weight monomial is
(the numbers on the horizontal line segments denote their
y
coordinates in ℤ × ℤ).
Fix a positive integer
n
. Let
S_{n}
be the group of permutations on
n
letters. Let
α
= (
α
_{1}
, . . . ,
α_{n}
),
β
= (
β
_{1}
, . . . ,
β_{n}
) ∈ ℤ
^{n}
be given with
α
_{1}
> . . . >
α_{n}
and
β
_{1}
> . . . >
β_{n}
. We define
Put
, and (−1)
^{p}
=
sgn
(
π
) for
p
∈ 𝒫 (
α
,
β
) with its associated permutation
π
∈
S_{n}
.
Example 3.4.
Let
n
= 4,
α
= (1, 0, −1, −2) and
β
= (4, 2, −1, −4). Then
with the associated permutation
A weight monomial of
p
is
and (−1)
^{p}
=
sgn
(
π
) = 1.
Let us define a map
as follows; for
p
= (
p
_{1}
, ...,
p_{n}
) ∈ 𝒫(
α
,
β
)

(1) Iffor all 1 ≤i≠j≤n, thenϕ(p) =p.

(2) Otherwise, we choose the largestisuch thatpihas an intersection pointwwithpjfor somei>j, and assume thatwis the first intersection point appearing inpifrom the bottom. Then we defineϕ(p) to be thentuple of paths obtained frompby replacing
Example 3.5.
Let
p
be as in Example 3.4. Then
By definition of
ϕ
, we can check that for
p
∈ 𝒫(
α
,
β
)

(1)ϕ(p) =pif and only ifphas no intersection point,

(2)ϕ2(p) =p,

(3)zϕ(p)=zp,

(4) (−1)ϕ(p)= −(−1)p.
We put
the set of fixed points in 𝒫(
α
,
β
) under
ϕ
, or the subset of
p
in 𝒫(
α
,
β
) with no intersection point.
For
, we define
where
δ
= (0, −1, . . . , −
n
+ 1) and
λ
+
δ
= (
λ
_{1}
,
λ
_{2}
− 1, . . . ,
λ_{n}
−
n
+ 1). For
k
∈ ℤ, we put 𝘚
_{k}
= 𝘚
_{(k)}
. Then we have the following JacobiTrudi formula for 𝘚
_{λ}
, which is an analogue of
[12]
for our zigzagshaped lattice paths.
Proposition 3.6.
For
,
we have
Proof
. Recall from (4) that for 1 ≤
i
,
j
≤
n
where we shift the
x
coordinates in
p
by −
j
+ 1. Thus
Since
ϕ
(
p
) ≠
p
for
p
∉ 𝒫
_{0}
(
δ
,
λ
+
δ
) and (−1)
^{ϕ(p)}
= −(−1)
^{p}
, we have
Also note that (−1)
^{p}
= 1 for
p
∈ 𝒫
_{0}
(
δ
,
λ
+
δ
). Therefore, we have
☐
3.2. Nonintersecting paths and Young tableaux
Let
α
= (
α
_{1}
, . . . ,
α_{n}
),
be such that
α
_{1}
> . . . >
α_{n}
,
β
_{1}
> . . . >
β_{n}
and
α_{i}
≤
β_{i}
for all
i
. Consider an
n
tuple
p
= (
p
_{1}
, . . . ,
p_{n}
) of nonintersecting (extended) lattice paths where
for 1 ≤
i
≤
n
. Note that
p_{i}
is a lattice path starting from a point (
α_{i}
, 0), which is a upper half of a lattice path defined in Definition 3.1. Put
δ
= (0, −1, . . . , −
n
+ 1). Choose
d
≥ 0 satisfying
If we put
µ
=
α
−
δ
+ (
d^{n}
) and
λ
=
β
−
δ
+ (
d^{n}
), then
λ
/
µ
is a skew Young diagram.
Now, associated to
p
, we define a tableau
T
of shape
λ
/
µ
with entries in ℤ
_{>0}
as follows. For 1 ≤
i
≤
n
with
α_{i}
<
β_{i}
and 1 ≤
j
≤
β_{i}
−
α_{i}
, we fill the box in the
i
th row and
j
th column of
λ
/
µ
with
k
if
The following lemma is wellknown
[7]
. But we give a detailed proof for the readers’ convenience.
Lemma 3.7.
Under the above assumptions, T is
ℤ
_{>0}

semistandard or a Young tableau of shape λ
/
µ
.
Proof
. Fix 1 ≤
i
≤
n
. Let
T_{i,j}
denote the
j
th (nonempty) entry of
T
(from the left) in the
i
th row (from the top) for 1 ≤
j
≤
β_{i}
−
α_{i}
.
It is clear that the entries of
T
in each row are weakly increasing from left to right since the
y
coordinates of each path
p_{i}
: (
α_{i}
, 0) → (
β_{i}
,∞) are weakly increasing from bottom to top. Hence it is enough to show that the entries of
T
in each column are strictly increasing from top to bottom.
Fix 1 ≤
i
<
n
. Suppose first that
α_{i}
−
α
_{i+1}
=
ℓ
≥ 1. Then
µ_{i}
−
µ
_{i+1}
= {
α_{i}
−(−
i
+ 1) +
d
} − {
α_{i}
−(−
i
) +
d
} = (
α_{i}
−
α
_{i+1}
)−1 =
ℓ
−1. This implies that
T_{i,j}
and
T
_{i+1,j+(ℓ−1)}
are in the same column in
T
for all
j
such that
T_{i,j}
and
T
_{i+1,j+(ℓ−1)}
are nonempty. The
j
th and (
j
+
ℓ
− 1)th horizontal line segments of
p_{i}
and
p
_{i+1}
are given by
respectively, where
k
=
T_{i,j}
and
k'
=
T
_{i+1,j+(ℓ−1)}
. If
k
≥
k'
, then the paths
p_{i}
and
p
_{i+1}
necessarily have an intersection point, which is a contradiction. Therefore,
T_{i,j}
<
T
_{i+1,j+(ℓ−1)}
. ☐
Note that the shape of
T
does not depend on the choice of
d
, and the correspondence
p
↦
T
gives a bijection between the set of nonintersecting paths satisfying (5) and
SST
_{ℤ>0}
(
λ
/
µ
).
Example 3.8.
Consider a quadruple of nonintersecting paths
p
= (
p
_{1}
,
p
_{2}
,
p
_{3}
,
p
_{4}
) with
α
= (−1, −3, −5, −6) and
β
= (3, 1, −2, −5)
Then the associated Young tableau is
Now, consider parabolically semistandard tableaux, where 𝓐 = ℤ
_{>0}
= { 1 < 2 < 3 < . . . } and 𝓑 = ℤ
_{<0}
= { −1 < −2 < −3 < . . . } with all entries even. Note that the linear ordering on 𝓑 is a reverse ordering of the usual one. Then we have
Proposition 3.9.
For
,
there exists a bijection
Proof
. Let
p
= (
p
_{1}
, . . . ,
p_{n}
) ∈ 𝒫
_{0}
(
δ, λ
+
δ
) be given with
for some
γ_{i}
∈ ℤ (1 ≤
i
≤
n
). Then we put
, where
, an upper half of
p_{i}
with the vertices having nonnegative second components, and put
, where
, the lower half of
p_{i}
. .
Choose
d
≥ 0 such that
γ
−
δ
+ (
d^{n}
),
. First, as in Lemma 3.7, we may associate a Young tableau
T
^{+}
of shape (
λ
+ (
d^{n}
))/
µ
where
µ
=
γ
−
δ
+ (
d^{n}
).
Let
, where
is obtained by reversing the order of the vertices in
and changing the sign of their second components. By the same argument, we may associate a Young tableau of shape (
d^{n}
)/
µ
, and then replace an entry
k
with −
k
once again to get a ℤ
_{<0}
semistandard tableau
T
^{−}
of (
d^{n}
)/
µ
.
We define a map
ψ
: 𝒫
_{0}
(
δ, λ
+
δ
) →
SST
_{ℤ>0/ℤ<0}
(
λ
) by
ψ
(
p
) = (
T
^{+}
,
T
^{−}
). Since the correspondence
p
↦ (
T
^{+}
,
T
^{−}
) is reversible,
ψ
is a bijection. ☐
Remark 3.10.
The bijection
ψ
in Proposition 3.9 preserves weight in the following sense: If (
T
^{+}
,
T
^{−}
) =
ψ
(
p
) for
p
∈ 𝒫
_{0}
(
δ, λ
+
δ
), then
, where we assume that
z_{k}
=
x_{k}
and
for
k
≥ 1.
Example 3.11.
Let
p
∈ 𝒫
_{0}
(
δ, λ
+
δ
) be a 4tuple of lattice paths with
δ
= (0, −1, −2, −3) and
λ
+
δ
= (3, 1, −2, −5) as follows.
Then
Now, we are in a position to prove our main theorem.
Theorem 3.12.
For
,
we have
Proof
. Let us assume that
z_{k}
=
x_{k}
and
for
k
≥ 1. Then we have
for
l
∈ ℤ. So by Proposition 3.6, we have
On the other hand, by Proposition 3.9 and Remark 3.10 we have
Combining (6) and (7), we obtain
This completes the proof. ☐
3.3. General cases for
𝓐
and
𝓑 In this subsection, we prove that Theorem 3.12 can be naturally extended to the case of
, where 𝓐 = ℤ
_{>0}
= { 1 < 2 < 3 < . . . } and 𝓑 = ℤ
_{<0}
= { −1 < −2 < −3 < . . . } with arbitrary ℤ
_{2}
gradings.
For this, we consider a lattice path
p
=
v
_{1}
...
v_{r}
of points
v
_{1}
, ...,
v_{r}
in ℤ × ℤ with
v_{i}
= (
s_{i}
,
t_{i}
) satisfying the following conditions:

(1)t1< 0

(2) ifti≠ 0 andti∈ 𝓐0⨆ 𝓑0, then


(3) ifti≠ 0 andti∈ 𝓐1⨆ 𝓑1, then


(4) ifti= 0, thenvi+1−vi= (0, 1).
We may define the notion of an extended path in the same way as in Section 3.1, and accordingly 𝒫(
α
,
β
), the involution
ϕ
, and 𝒫
_{0}
(
α
,
β
). For an (extended) path
p
and
z
= {
z_{i}

i
∈ ℤ
^{×}
} the set of formal commuting variables, we put
where
p
= (
s
_{1}
, −∞)
v
_{1}
. . .
v_{r}
(
s_{r}
,∞) with
v_{i}
= (
s_{i}
,
t_{i}
) for 1 ≤
i
≤
r
. Then we define
and we have
by the same arguments as in Proposition 3.6.
Example 3.13.
Suppose that ℤ
_{2}
gradings on 𝓐 and 𝓑 are given by
For a lattice path
its weight monomial is
(the numbers on the horizontal or the diagonal denote their
y
coordinates in ℤ × ℤ).
Now, for
, there is also a weightpreserving bijection from 𝒫
_{0}
(
δ, λ
+
δ
) to
SST
_{𝓐/𝓑}
(
λ
) (see Lemma 3.7 and Proposition 3.9).
Example 3.14.
We assume that 𝓐 and 𝓑 are as in (9). Let
p
∈ 𝒫
_{0}
(
δ, λ
+
δ
) be a 4tuple of lattice paths with
δ
= (0, −1, −2, −3) and
λ
+
δ
= (3, 1, −2, −5) as follows.
Then it corresponds to
Therefore, combining with (8), we obtain the JacobiTrudi type formula for
.
Theorem 3.15.
For
,
we have
Remark 3.16.
One can also prove Theorem 3.15 when 𝓐 and 𝓑 are arbitrary two disjoint linearly ordered ℤ
_{2}
graded sets, by slightly modifying the notion of extended paths.
2010 Mathematics Subject Classification. 05E05,17B67.
Acknowledgements
The author would like to thank J.H. Kwon for many helpful advices.
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