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. 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 well-known 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 Kac-Moody 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 well-known one is the Jacobi-Trudi 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 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 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 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 S_λ 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 S_λ . 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 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 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 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 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 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 𝓐-semistandard 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 𝓐-semistandard 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 ⊂ ( dⁿ ), µ ⊂ λ +( dⁿ ). 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 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 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 multiplicity-free 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 𝔤-module 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). 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_rv _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_iv _i+1 denote the line segment joining v_i and v _i+1 , where we understand v ₀ v ₁ (resp. v_rv _r+1 ) as an half-infinite 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 ) 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 ) 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 Jacobi-Trudi formula for 𝘚 _λ , which is an analogue of [12] for our zigzag-shaped 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. Non-intersecting 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 non-intersecting (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 + ( 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 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 well-known [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 (non-empty) 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 non-empty. 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 non-intersecting paths satisfying (5) and SST _ℤ>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) 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 non-negative second components, and put , where , the lower half of p_i . . Choose d ≥ 0 such that γ − δ + ( dⁿ ), . First, as in Lemma 3.7, we may associate a Young tableau T ⁺ of shape ( λ + ( dⁿ ))/ µ where µ = γ − δ + ( dⁿ ). 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 ( 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 _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 4-tuple 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: 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 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. Then it corresponds to Therefore, combining with (8), we obtain the Jacobi-Trudi 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.