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JACOBI-TRUDI TYPE FORMULA FOR PARABOLICALLY SEMISTANDARD TABLEAUX
JACOBI-TRUDI TYPE FORMULA FOR PARABOLICALLY SEMISTANDARD TABLEAUX
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2015. Aug, 22(3): 245-261
Copyright © 2015, Korean Society of Mathematical Education
  • Received : February 23, 2015
  • Accepted : July 27, 2015
  • Published : August 31, 2015
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Jee-Hye Kim

Abstract
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 Jacobi-Trudi type formula for the character of parabolically semistandard tableaux of a given generalised partition shape by using non-intersecting lattice paths.
Keywords
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 , . . . , xn be mutually commuting n variables. Let sλ ( x 1 , . . . , xn ) be the Schur polynomial corresponding to a partition λ = ( λ 1 , . . . , λn ). It is well-known that sλ ( x 1 , . . . , xn ) is the character of a complex irreducible polynomial representation of the general linear group GLn (ℂ) whose highest weight corresponds to λ . There are several equivalent definitions of sλ ( x 1 , . . . , xn ). One is the celebrated Weyl character formula, which has been extended to the case of a symmetrizable Kac-Moody algebra [9] . There is a combinatorial formula, where sλ ( x 1 , . . . , xn ) is given as the weight generating function of the set of Young tableaux of shape λ . Another well-known one is the Jacobi-Trudi formula
PPT Slide
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where hk ( x 1 , . . . , xn ) 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 non-intersecting 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 Sλ 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, GLn (ℂ)-duality on various Fock spaces. The character Sλ includes a usual Schur polynomial as a special case, and it also has a Weyl-Kac type character formula, and a Jacobi-Trudi type formula (see also [13] ).
The goal of this paper is to show a Jacobi-Trudi type formula for Sλ by using non-intersecting lattice paths. Since a parabolically semistandard tableau is roughly speaking a pair ( S , T ) of skew-shaped Young tableaux with a common inner shape and each component corresponds to an n -tuple of non-intersecting lattice paths, the pair ( S , T ) corresponds to an n -tuple of non-intersecting zigzag-shaped lattice paths which is obtained by gluing non-intersecting paths associated to S and T . This is our key observation. Then we apply the arguments similar to [7] to obtain a Jacobi-Trudi type formula for Sλ .
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 non-negative integers λ = ( λ 1 , λ 2 , . . .) such that Σ i≥1 λi is finite. We say that λ is a partition of n if Σ i≥1 λi = n and denote by ( λ ) the number of positive entries of λ . Let
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be the set of all partitions, and put
PPT Slide
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for n ≥ 1.
A Young diagram is a collection of boxes arranged in left-justified 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 λi 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
PPT Slide
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Let
PPT Slide
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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)
PPT Slide
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is a Young tableau of shape λ .
For
PPT Slide
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with λ µ (that is, λi µi 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),
PPT Slide
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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
PPT Slide
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, where mi is the number of times i occurs in T . For T in Example 2.2, we have
PPT Slide
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. Let sλ (x) = Σ T x T be the Schur function corresponding to
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, where the sum is over all Young tableaux T of sh ( T ) = λ . For k ≥ 0, let hk ( x ) = s (k) (x), which is called the k th complete symmetric function . For
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, we put hµ (x) = h µ1 (x) h µ2 (x). . ..
There is another well-known equivalent definition of a Schur function called the Jacobi-Trudi formula , which expresses a Schur function as a determinant, and hence as a linear combination of hµ (x)’s for
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(cf. [6] ).
Theorem 2.4. For
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with ( λ ) ≤ n ,
PPT Slide
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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 𝓐-semistandard tableaux of shape λ / µ . We set
PPT Slide
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. Let x 𝓐 = { xa | a ∈ 𝓐 } be a set of formal commuting variables indexed by 𝓐. For T SST𝓐 ( λ / µ ), put
PPT Slide
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, where ma is the number of occurrences of a in T . We define the character of SST 𝓐 ( λ / µ ) to be
PPT Slide
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.
We assume that ℤ >0 is given with a usual linear ordering and all entries even. When 𝓐 = ℤ >0 , an 𝓐-semistandard tableau is a (skew) Young tableau, and sλ (x 𝓐 ) is the Schur function associated to
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.
Let
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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
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corresponds to
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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
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, a parabolically semistandard tableau of shape λ (with respect to (𝓐, 𝓑)) is a pair of tableaux ( T + , T ) such that
PPT Slide
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for some integer d ≥ 0 and
PPT Slide
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satisfying (1)
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, and (2) µ ⊂ ( dn ), µ λ +( dn ). We denote by SST𝓐/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 𝓐-semistandard tableau and 𝓑-semistandard 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
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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
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, we define the character of SST𝓐/𝓑 ( λ ) to be
PPT Slide
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We put
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. For k ∈ ℤ, we put
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.
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 { vc | c ∈ 𝓒 }, and let 𝔤𝔩( V 𝓒 ) be the general linear Lie superalgebra spanned by Ecc' for c , c' ∈ 𝓒. Here Ecc' 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
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be the super symmetric algebra generated by
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, where
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is the restricted dual space of V 𝓑 . One can define a semisimple action of 𝔤 on
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, and a semisimple action of GLn (ℂ) on
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for n ≥ 1 so that we have the following multiplicity-free decomposition as a (𝔤, GLn (ℂ))-module,
PPT Slide
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for a subset Hn of
PPT Slide
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, where Ln ( λ ) is the irreducible GLn (ℂ)-module with highest weight λ Hn , and L ( λ ) is an irreducible 𝔤-module corresponding to Ln ( 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
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on L ( λ ) for λ Hn . 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
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as a (𝔤, GLn (ℂ))- module , that is ,
PPT Slide
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, and the irreducible character ch L ( λ ) is given by
PPT Slide
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for
PPT Slide
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.
Recall that when 𝓐 is finite with 𝓐 = 𝓐 0 or 𝓐 1 and
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, the decomposition in Theorem 2.8 is the classical ( GL (ℂ), GLn (ℂ))-Howe duality on the symmetric algebra or exterior algebra generated by ℂ ⊗ ℂ n , where = |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. JACOBI-TRUDI FORMULA
- 3.1. Lattice paths
Definition 3.1. A lattice path is a sequence
PPT Slide
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of points v 1 , ..., vr in ℤ × ℤ with vi = ( ai , bi ) such that b 1 < 0 < br , and
PPT Slide
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Example 3.2. The following path
PPT Slide
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is the lattice path
PPT Slide
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We denote by 𝒫 the set of lattice paths. Let p = v 1 ... vr ∈ 𝒫 be given with vi = ( ai , bi ) for 1 ≤ i r . We often identify p with its extended lattice path v 0 v 1 ... vrv r+1 , where v 0 = ( a 1 , −∞) and v r+1 = ( ar , ∞). Here we regard ( a 1 , −∞) as a point below ( a 1 , y ) for all y b 1 , and ( ar , ∞) as a point above ( ar , y ) for all y br . We also write p : v 0 v r+1 . For 0 ≤ i r , let viv i+1 denote the line segment joining vi and v i+1 , where we understand v 0 v 1 (resp. vrv r+1 ) as an half-infinite line joining ( a 1 , b 1 ) and ( a 1 , −∞) (resp. ( ar , br ) and ( ar , ∞)). Let z = { zi | i ∈ ℤ × } be a set of formal commuting variables, where ℤ × = ℤ \ {0}. We consider a weight monomial
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Example 3.3. For a lattice path
PPT Slide
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its weight monomial is
PPT Slide
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(the numbers on the horizontal line segments denote their y -coordinates in ℤ × ℤ).
Fix a positive integer n . Let Sn be the group of permutations on n letters. Let α = ( α 1 , . . . , αn ), β = ( β 1 , . . . , βn ) ∈ ℤ n be given with α 1 > . . . > αn and β 1 > . . . > βn . We define
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Put
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, and (−1) p = sgn ( π ) for p ∈ 𝒫 ( α , β ) with its associated permutation π Sn .
Example 3.4. Let n = 4, α = (1, 0, −1, −2) and β = (4, 2, −1, −4). Then
PPT Slide
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with the associated permutation
PPT Slide
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A weight monomial of p is
PPT Slide
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and (−1) p = sgn ( π ) = 1.
Let us define a map
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as follows; for p = ( p 1 , ..., pn ) ∈ 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 then-tuple of paths obtained frompby replacing
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Example 3.5. Let p be as in Example 3.4. Then
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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
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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
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, we define
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where δ = (0, −1, . . . , − n + 1) and λ + δ = ( λ 1 , λ 2 − 1, . . . , λn n + 1). For k ∈ ℤ, we put 𝘚 k = 𝘚 (k) . Then we have the following Jacobi-Trudi formula for 𝘚 λ , which is an analogue of [12] for our zigzag-shaped lattice paths.
Proposition 3.6. For
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, we have
PPT Slide
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Proof . Recall from (4) that for 1 ≤ i , j n
PPT Slide
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where we shift the x -coordinates in p by − j + 1. Thus
PPT Slide
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Since ϕ ( p ) ≠ p for p ∉ 𝒫 0 ( δ , λ + δ ) and (−1) ϕ(p) = −(−1) p , we have
PPT Slide
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Also note that (−1) p = 1 for p ∈ 𝒫 0 ( δ , λ + δ ). Therefore, we have
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                              ☐
3.2. Non-intersecting paths and Young tableaux Let α = ( α 1 , . . . , αn ),
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be such that α 1 > . . . > αn , β 1 > . . . > βn and αi βi for all i . Consider an n -tuple p = ( p 1 , . . . , pn ) of non-intersecting (extended) lattice paths where
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for 1 ≤ i n . Note that pi is a lattice path starting from a point ( αi , 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
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If we put µ = α δ + ( dn ) and λ = β δ + ( dn ), 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 αi < βi and 1 ≤ j βi αi , we fill the box in the i th row and j th column of λ / µ with k if
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The following lemma is well-known [7] . But we give a detailed proof for the readers’ convenience.
Lemma 3.7. Under the above assumptions, T is >0 - semistandard or a Young tableau of shape λ / µ .
Proof . Fix 1 ≤ i n . Let Ti,j denote the j th (non-empty) entry of T (from the left) in the i th row (from the top) for 1 ≤ j βi αi .
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 pi : ( αi , 0) → ( βi ,∞) 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 αi α i+1 = ≥ 1. Then µi µ i+1 = { αi −(− i + 1) + d } − { αi −(− i ) + d } = ( αi α i+1 )−1 = −1. This implies that Ti,j and T i+1,j+(−1) are in the same column in T for all j such that Ti,j and T i+1,j+(−1) are non-empty. The j th and ( j + − 1)th horizontal line segments of pi and p i+1 are given by
PPT Slide
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respectively, where k = Ti,j and k' = T i+1,j+(−1) . If k k' , then the paths pi and p i+1 necessarily have an intersection point, which is a contradiction. Therefore, Ti,j < T i+1,j+(−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 non-intersecting paths satisfying (5) and SST >0 ( λ / µ ).
Example 3.8. Consider a quadruple of non-intersecting paths p = ( p 1 , p 2 , p 3 , p 4 ) with α = (−1, −3, −5, −6) and β = (3, 1, −2, −5)
PPT Slide
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Then the associated Young tableau is
PPT Slide
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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
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, there exists a bijection
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Proof . Let p = ( p 1 , . . . , pn ) ∈ 𝒫 0 ( δ, λ + δ ) be given with
PPT Slide
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for some γi ∈ ℤ (1 ≤ i n ). Then we put
PPT Slide
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, where
PPT Slide
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, an upper half of pi with the vertices having non-negative second components, and put
PPT Slide
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, where
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, the lower half of pi . .
Choose d ≥ 0 such that γ δ + ( dn ),
PPT Slide
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. First, as in Lemma 3.7, we may associate a Young tableau T + of shape ( λ + ( dn ))/ µ where µ = γ δ + ( dn ).
Let
PPT Slide
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, where
PPT Slide
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is obtained by reversing the order of the vertices in
PPT Slide
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and changing the sign of their second components. By the same argument, we may associate a Young tableau of shape ( dn )/ µ , and then replace an entry k with − k once again to get a ℤ <0 -semistandard tableau T of ( dn )/ µ .
We define a map ψ : 𝒫 0 ( δ, λ + δ ) → SST >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
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, where we assume that zk = xk and
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for k ≥ 1.
Example 3.11. Let p ∈ 𝒫 0 ( δ, λ + δ ) be a 4-tuple of lattice paths with δ = (0, −1, −2, −3) and λ + δ = (3, 1, −2, −5) as follows.
PPT Slide
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Then
PPT Slide
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Now, we are in a position to prove our main theorem.
Theorem 3.12. For
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, we have
PPT Slide
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Proof . Let us assume that zk = xk and
PPT Slide
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for k ≥ 1. Then we have
PPT Slide
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for l ∈ ℤ. So by Proposition 3.6, we have
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On the other hand, by Proposition 3.9 and Remark 3.10 we have
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Combining (6) and (7), we obtain
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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
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, 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 ... vr of points v 1 , ..., vr in ℤ × ℤ with vi = ( si , ti ) 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 = { zi | i ∈ ℤ × } the set of formal commuting variables, we put
PPT Slide
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where p = ( s 1 , −∞) v 1 . . . vr ( sr ,∞) with vi = ( si , ti ) for 1 ≤ i r . Then we define
PPT Slide
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and we have
PPT Slide
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by the same arguments as in Proposition 3.6.
Example 3.13. Suppose that ℤ 2 -gradings on 𝓐 and 𝓑 are given by
PPT Slide
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For a lattice path
PPT Slide
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its weight monomial is
PPT Slide
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(the numbers on the horizontal or the diagonal denote their y -coordinates in ℤ × ℤ).
Now, for
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, there is also a weight-preserving 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 4-tuple of lattice paths with δ = (0, −1, −2, −3) and λ + δ = (3, 1, −2, −5) as follows.
PPT Slide
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Then it corresponds to
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Therefore, combining with (8), we obtain the Jacobi-Trudi type formula for
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.
Theorem 3.15. For
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, we have
PPT Slide
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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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