Determinantal expression and recursion for Jack polynomials

Luc Lapointe Centre de recherches mathématiques Université de Montréal, C.P. 6128, succ. Centre-Ville, Montréal, Québec H3C 3J7, Canada

                    lapointe@crm.umontreal.ca

A. Lascoux Institut Gaspard Monge, Université de Marne-la-Vallée 5 Bd Descartes, Champs sur Marne 77454 Marne La Vallée, Cedex, FRANCE

                    Alain.Lascoux@univ-mlv.fr

J. Morse Department of Mathematics University of Pennsylvania 209 South 33rd Street, Philadelphia, PA 19103, USA

                      morsej@math.upenn.edu

Submitted: November 3, 1999; Accepted: November 22, 1999.

AMS Subject Classification: 05E05.

Abstract

We describe matrices whose determinants are the Jack polynomials expanded in terms of the monomial basis. The top row of such a matrix is a list of monomial functions, the entries of the sub-diagonal are of the form −(rα + s), with r and s ∈ ℕ^{+}, the entries above the sub-diagonal are nonnegative integers, and below all entries are 0. The quasi-triangular nature of these matrices gives a recursion for the Jack polynomials allowing for efficient computation. A specialization of these results yields a determinantal formula for the Schur functions and a recursion for the Kostka numbers.

1 Introduction

The Jack polynomials J_{λ}[x_{1}, …, x_{N}; α] form a basis for the space of N-variable symmetric polynomials. Here we give a matrix of which the determinant is J_{λ}[x; α] expanded in terms of the monomial basis. The top row of this matrix is a list of monomial functions, the entries of the sub-diagonal are of the form −(rα + s), with r and s ∈ ℕ^{+}, the entries above the sub-diagonal are non-negative integers, and below all entries are 0. The quasi-triangular nature of this matrix gives a simple recursion for the Jack polynomials allowing for their rapid computation. The result here is a transformed specialization of the matrix expressing Macdonald polynomials given in [2]. However, we give a self-contained derivation of the matrix for Jack polynomials. Since the Schur functions s_{λ}[x] are the specialization α = 1 in J_{λ}[x; α], we obtain a matrix of which the determinant gives s_{λ}[x]. A by-product of this result is a recursion for the Kostka numbers, the expansion coefficients of the Schur functions in terms of the monomial basis.

Partitions are weakly decreasing sequences of non-negative integers. We use the dominance order on partitions, defined μ ≤ λ ⟺ μ_{1} + ⋯ + μ_{i} ≤ λ_{1} + ⋯ + λ_{i} ∀i. The number of non-zero parts of a partition λ is denoted ℓ(λ). The Jack polynomials can be defined up to normalization by the conditions

\begin{matrix}(i) & J_{λ} = ∑_{μ≤λ} v_{λμ}m_{μ} ,  with v_{λλ} ≠ 0 , \\ (ii) & HJ_{λ} = [∑_{i=1}^{N} (\frac{α}{2}λ_{i}^{2} + \frac{1}{2}(N + 1 − 2i)λ_{i})] J_{λ} , \end{matrix}  (1)

where H is the Hamiltonian of the Calogero-Sutherland model [7] defined

H = \frac{α}{2} ∑_{i=1}^{N} (x_{i} \frac{∂}{∂x_{i}} )^{2} + \frac{1}{2} ∑_{i<j} (\frac{x_{i} + x_{j}}{x_{i} − x_{j}} ) (x_{i} \frac{∂}{∂x_{i}} − x_{j} \frac{∂}{∂x_{j}} ) .  (2)

A composition β = (β_{1}, …, β_{n}) is a vector of non-negative integral components and the partition rearrangement of β is denoted β^{∗}. The raising operator R_{ij}^{ℓ} acts on compositions by R_{ij}^{ℓ}β = (β_{1}, …, β_{i} − ℓ, …, β_{j} + ℓ, …, β_{n}), for any i < j. We will use n(k) to denote the number of occurrences of k in μ. This given, we use the following theorem [5] :

Theorem 1. Given a partition λ, we have

H m_{λ} = [∑_{i=1}^{ℓ(λ)} (\frac{α}{2}λ_{i}^{2} + \frac{1}{2}(N + 1 − 2i)λ_{i})] m_{λ} + ∑_{μ<λ} C_{λμ} m_{μ} ,  (3)

where if there exists some i < j, and 1 ≤ ℓ ≤ ⌊\frac{λ_{i}−λ_{j}}{2} ⌋ such that (R_{ij}^{ℓ}λ)^{∗} = μ, then

C_{λμ} = \begin{cases}(λ_{i} − λ_{j})\binom{n(μ_{i})}{2} & if μ_{i} = μ_{j} \\ (λ_{i} − λ_{j})n(μ_{i})n(μ_{j}) & if μ_{i} ≠ μ_{j} \end{cases}  (4)

and otherwise C_{λμ} = 0.

\underline{Example 1}: with N = 5,

\begin{matrix}H & m_{4} = (8 + 8α)m_{4} + 4 m_{3,1} + 4 m_{2,2} H m_{2,1,1} = (5 + 3α) m_{3,1} + 12 m_{1,1,1,1} \\ H & m_{3,1} = (7 + 5α) m_{3,1} + 2 m_{2,2} + 6 m_{2,1,1} H m_{1,1,1,1} = (2 + 2α) m_{1,1,1,1} \\ H & m_{2,2} = (6 + 4α) m_{2,2} + 2 m_{2,1,1} \end{matrix}

2 Determinant for the Jack polynomials

We can obtain non-vanishing determinants which are eigenfunctions of the Hamiltonian H by using the triangular action of H on the monomial basis.

Theorem 2. If μ^{(1)}, μ^{(2)}, …, μ^{(n)} = μ is a linear ordering of all partitions ≤ μ, then the Jack polynomial J_{μ} is proportional to the following determinant;

J_{μ} ≐ |\begin{matrix}m_{μ^{(1)}} & m_{μ^{(2)}} & … & … & m_{μ^{(n−1)}} & m_{μ^{(n)}} \\ d_{μ^{(1)}} − d_{μ^{(n)}} & C_{μ^{(2)}μ^{(1)}} & … & … & C_{μ^{(n−1)}μ^{(1)}} & C_{μ^{(n)}μ^{(1)}} \\ 0 & d_{μ^{(2)}} − d_{μ^{(n)}} & & & … & C_{μ^{(n)}μ^{(2)}} \\ ⋮ & 0 & ... & & & ⋮ \\ ⋮ & ⋮ & ... & ... & & ⋮ \\ 0 & 0 & … & 0 & d_{μ^{(n−1)}} − d_{μ^{(n)}} & C_{μ^{(n)}μ^{(n−1)}} \end{matrix} |  (5)

where d_{λ} denotes the eigenvalue in 1(ii) and C_{μ^{(i)}μ^{(j)}} is defined by (4).

Note that in the case μ = (a), the matrix J_{a} contains all possible C_{μ^{(i)}μ^{(j)}}. Therefore, the matrices corresponding to J_{1}, J_{2}, … determine the entries off the subdiagonal for all other matrices. Further, the sub-diagonal entries d_{μ^{(i)}} − d_{μ^{(n)}} do not depend on the number of variables N, when N ≥ ℓ(μ), since for any partitions μ and λ where ℓ(μ) ≥ ℓ(λ),

d_{μ} − d_{λ} = ∑_{i=1}^{ℓ(μ)} (\frac{α}{2}(μ_{i}^{2} − λ_{i}^{2}) − i(μ_{i} − λ_{i})) .  (6)

It is also easily checked that if μ < λ, then d_{μ} − d_{λ} = −(r + sα) for some r, s ∈ ℕ^{+}, showing that the sub-diagonal entries are of this form.

\underline{Example 2}: The entries of the matrix for J_{4} are obtained using the action of H on m_{μ^{(i)}}, where μ^{(1)} =(1, 1, 1, 1), μ^{(2)} =(2, 1, 1), μ^{(3)} =(2, 2), μ^{(4)} =(3, 1), μ^{(5)} =(4), given in Example 1;

J_{4} ≐ |\begin{matrix}m_{1,1,1,1} & m_{2,1,1} & m_{2,2} & m_{3,1} & m_{4} \\ −6 − 6α & 12 & 0 & 0 & 0 \\ 0 & −3 − 5α & 2 & 6 & 0 \\ 0 & 0 & −2 − 4α & 2 & 4 \\ 0 & 0 & 0 & −1 − 3α & 4 \end{matrix}|  (7)

We also obtain a determinantal expression for the Schur functions in terms of monomials using Theorem 2 since s_{λ}[x] is the specialization J_{λ}[x; 1].

Corollary 3. Given a partition μ, the specialization α=1 in the determinant (5) is proportional to the Schur function s_{μ}.

\underline{Example 3}: s_{4} can be obtained by specializing α = 1 in the matrix (7).

s_{4} ≐ |\begin{matrix}m_{1,1,1,1} & m_{2,1,1} & m_{2,2} & m_{3,1} & m_{4} \\ −12 & 12 & 0 & 0 & 0 \\ 0 & −8 & 2 & 6 & 0 \\ 0 & 0 & −6 & 2 & 4 \\ 0 & 0 & 0 & −4 & 4 \end{matrix}|  (8)

Proof of Theorem 2. We have from (6) that d_{λ} ≠ d_{μ} for λ < μ implying that the sub-diagonal entries, d_{μ^{(i)}} −d_{μ}, of determinantal expression (5) are non-zero. Since the coefficient of m_{μ^{(n)}} = m_{μ} is the product of the sub-diagonal elements, this coefficient does not vanish and by the construction of J_{μ}, Property 1(i) is satisfied. It thus suffices to check that (H − d_{μ}) J_{μ} = 0. Since H acts non-trivially only on the first row of the determinant J_{μ}, the first row of (H − d_{μ}) J_{μ} is obtained from Theorem 1 and expression (5) gives rows 2, …, n.

(H − d_{μ})J_{μ} = | … d_{μ^{(j)}}m_{μ^{(j)}} + ∑_{i<j} C_{μ^{(j)}μ^{(i)}}m_{μ^{(i)}} − d_{μ}m_{μ^{(j)}} …; …  C_{μ^{(j)}μ^{(1)}}  …; ⋮ ; …  C_{μ^{(j)}μ^{(j−1)}}  …; …  d_{μ^{(j)}} − d_{μ}  …; …  0  …; ⋮ |

m_{μ^{(n)}} appears only in the first row, column n, with coefficient d_{μ}−d_{μ} = 0. Further, we have that the first row is the linear combination: m_{μ^{(1)}}row_{2} + m_{μ^{(2)}}row_{3} + ⋯ + m_{μ^{(n−1)}}row_{n}, and thus the determinant must vanish.

3 Recursion for quasi-triangular matrices

We use the determinantal expressions for Jack and Schur polynomials to obtain recursive formulas. First we will give a recursion for J_{λ}[x; α] providing an efficient method for computing the Jack polynomials and we will finish our note by giving a recursive definition for the Kostka numbers. These results follow from a general property of quasi-triangular determinants [3, 8];

Property 4. Any quasi-triangular determinant of the form

D = |\begin{matrix}b_{1} & b_{2} & ⋯ & b_{n−1} & b_{n} \\ −a_{21} & a_{22} & ⋯ & a_{2,n−1} & a_{2,n} \\ 0 & −a_{32} & ⋯ & a_{3,n−1} & a_{3,n} \\ ⋮ & ... & ... & & ⋮ \\ 0 & 0 & 0 & −a_{n,n−1} & a_{n,n} \end{matrix}|  (9)

has the expansion D = ∑_{i=1}^{n} c_{i}b_{i}, where c_{n} = a_{21}a_{32} ⋯ a_{n,n−1} and

c_{i} = \frac{1}{a_{i+1,i}} ∑_{j=i+1}^{n} a_{i+1,j} c_{j} for all i ∈ \{1, 2, …, n − 1\} .  (10)

Proof. Given linearly independent n-vectors b and a^{(i)}, i ∈ {2, …, n}, let c be a vector orthogonal to a^{(2)}, …, a^{(n)}. This implies that the determinant of a matrix with row vectors b, a^{(2)}, …, a^{(n)} is the scalar product (c, b) = ∑_{i=1}^{n} c_{i}b_{i}, up to a normalization. In the particular case of matrices with the form given in (9), that is with a^{(i)} = (0, …, 0, −a_{i,i−1}, a_{i,i}, …, a_{i,n}), i ∈ {2, …, n}, we can see that (c, a^{(i)}) = 0 if the components of c satisfy recursion (10). Since we have immediately that the coefficient of b_{n} in (9) is c_{n} = a_{21} ⋯ a_{n,n−1}, ensuring that ∑_{i=1}^{n} c_{i}b_{i} is properly normalized, Property 4 is thus proven.

Notice that we can freely multiply the rows of matrix (9) by non-zero constants and still preserve recursion (10). This implies that to obtain a matrix proportional to determinant (9), one would simply multiply the value of c_{n} by the proportionality constant.

Since the determinantal expression (5) for the Jack polynomials is of the form that appears in Property 4, we may compute the Jack polynomials, in any normalization, using recursion (10).

\underline{Example:} Recall that v_{(4);(4)} = (1+α)(1+2α)(1+3α) in the normalization associated to the positivity of Jack polynomials [4, 6, 1]. Thus, using

J_{4} ≐ |\begin{matrix}m_{1,1,1,1} & m_{2,1,1} & m_{2,2} & m_{3,1} & m_{4} \\ −6 − 6α & 12 & 0 & 0 & 0 \\ 0 & −3 − 5α & 2 & 6 & 0 \\ 0 & 0 & −2 − 4α & 2 & 4 \\ 0 & 0 & 0 & −1 − 3α & 4 \end{matrix}| ,  (11)

and initial condition c_{5} = (1 + α)(1 + 2α)(1 + 3α), we get

\begin{aligned}c_{4} & = \frac{1}{1 + 3α} (4c_{5}) = 4(1 + α)(1 + 2α) , c_{2} = \frac{1}{3 + 5α} (2c_{3} + 6c_{4}) = 12(1 + α) , \\ c_{3} & = \frac{1}{2 + 4α} (2c_{4} + 4c_{5}) = 6(1 + α)^{2} , c_{1} = \frac{1}{6 + 6α} (12c_{2}) = 24 , \end{aligned} (12)

which gives that

\begin{aligned}J_{4} = & (1 + α)(1 + 2α)(1 + 3α)m_{4} + 4(1 + α)(1 + 2α)m_{3,1} \\ & + 6(1 + α)^{2}m_{2,2} + 12(1 + α)m_{2,1,1} + 24m_{1,1,1,1} . \end{aligned}  (13)

If we let μ^{(1)}, μ^{(2)}, …, μ^{(n)} = μ be a linear ordering of all partitions ≤ μ and recall that the Kostka numbers are the coefficients K_{μμ^{(i)}} in

s_{μ}[x] = ∑_{i=1}^{n} K_{μμ^{(i)}}m_{μ^{(i)}}[x] where K_{μμ^{(n)}} = K_{μμ} = 1 ,  (14)

we can use Property 4 to obtain a recursion for the K_{μμ^{(i)}}.

Corollary 5. Let μ^{(1)}, μ^{(2)}, …, μ^{(n)} = μ be a linear ordering of all partitions ≤ μ. K_{μμ^{(i)}} is defined recursively, with initial condition K_{μμ} = 1, by

K_{μμ^{(i)}} = \frac{1}{g_{μ^{(i)}} − g_{μ}} ∑_{j=i+1}^{n} C_{μ^{(i+1)}μ^{(j)}}K_{μμ^{(j)}} for all i ∈ \{1, 2, …, n − 1\},  (15)

where C_{μ^{(i+1)}μ^{(j)}} is given in (4) and where g_{μ^{(i)}} − g_{μ} is the specialization α = 1 of d_{μ^{(i)}} − d_{μ} introduced in (4).

Acknowledgments. This work was completed while L. Lapointe held a NSERC postdoctoral fellowship at the University of California at San Diego.

References

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