The symmetric functions catalog

An overview of symmetric functions and related topics

2021-07-21

Jack P polynomials

The Jack polynomials are a family of symmetric functions which extends the Schur polynomials. They were introduced by H. Jack in [Jac70]. They are indexed by integer partitions and constitute a basis for the space of symmetric functions. For an overview, see [Mac95] and [Sta89].

The Jack polynomials can be generalized to the shifted Jack polynomials, Jack interpolation polynomials and Macdonald $P$ polynomials.

Deformed Hall inner product

Let $\langle \cdot, \cdot \rangle_a$ be the inner product on symmetric functions such that $\langle \powerSum_\lambda, \powerSum_\mu \rangle_a = \delta_{\lambda\mu} a^{\length(\lambda)} z_\lambda.$ Then the family $\jackP_\lambda(x;a)$ is the unique family that satisfies:

  • Orthogonality: $\langle \jackP_\lambda, \jackP_\mu \rangle_a = 0$ whenever $\lambda \neq \mu.$
  • Triangularity: $\jackP_\lambda = \sum_{\mu \lt_d \lambda } c_{\lambda \mu} \monomial_\mu.$
  • Normalization: $[\monomial_{1^n}]\jackP_\lambda = 1.$

These symmetric functions have coefficients which are rational functions in $a.$

RSSYT formula

The Jack polynomial $\jackP_\mu(\xvec;a)$ in $n$ variables may be defined as

\[ \jackP_\mu(\xvec;a) = \sum_{T \in \textrm{RSSYT}(\mu)} \psi_T(a) \prod_{s \in \mu} x_{T(s)}, \]

where the sum is taken over all reverse tableau with entries in $[n]$ and shape $\mu.$ Here, $\psi_T(a)$ is the rational function defined in [Mac95] as

\begin{equation*} \psi_T(a) = \prod_{i=1}^n \psi_{\rho^i/\rho^{i-1}}(a) \end{equation*}

where $\rho^i/\rho^{i-1}$ defines the skew-shape in $T$ with content $i$ ($\rho^0 = \emptyset$) and

\begin{equation*} \psi_{\lambda/\mu}(a) = \prod_{s \in R_{\lambda/\mu} \setminus C_{\lambda/\mu} } \frac{ (a \cdot\arm_\lambda(s) + \leg_\lambda(s) + a)(a \cdot \arm_\mu(s) + \leg_\mu(s) + 1) }{ (a\cdot \arm_\lambda(s) + \leg_\lambda(s) + 1)(a \cdot\arm_\mu(s) + \leg_\mu(s) + a) }. \end{equation*}

Here, $R_{\lambda/\mu}$ denotes the set of boxes in a row that intersects the shape $\lambda/\mu.$ The set of boxes $C_{\lambda/\mu}$ is defined in a similar manner for columns.

This formula generalizes to the super Jack polynomials, see [SV05].

Example (Example of $\psi_{\lambda/\mu}(a)$ ).

If $\lambda/\mu = (7,5,3,2)/(5,4,2,2),$ then the product for computing $\psi_{\lambda/\mu}(a)$ is taken over all boxes marked with a dot in

$\cdot$$\cdot$ $\cdot$ $\times$$\times$
$\cdot$$\cdot$ $\cdot$$\times$  
$\cdot$$\cdot$$\times$    
       

From this definition, it is evident that $\jackP_\mu(\xvec;1)$ is the Schur polynomial $\schurS_\mu(\xvec).$

Recursive formula

There is an efficient recursion for computing the Kostka coefficients $K_{\lambda\mu}(\alpha),$ appearing in the expansion $\jackP_\lambda = \sum_{\mu} K_{\lambda\mu}(\alpha) \monomial_\lambda.$ This recursion be found in [p.327, Mac95] where he uses the notation $u_{\lambda\mu}$ for these coefficients. See also [LLM99Rob00], and [Prop. 2.16, DES07] where this formula appears. Generalizations to other root systems can be found in [DLM04b].

Jack J polynomials

The integral form Jack polynomals are defined as the unique family satisfying the following relations:

  • Orthogonality: $\langle \jackJ_\lambda, \jackJ_\mu \rangle_a = 0$ whenever $\lambda \neq \mu.$
  • Triangularity: $\jackJ_\lambda = \sum_{\mu \lt_d \lambda } c_{\lambda \mu} \monomial_\mu.$
  • Normalization: $[\monomial_{1^n}]\jackJ_\lambda = n!.$

These symmetric functions have coefficients in $\setN[a],$ see the combinatorial formula below.

It is convenient to introduce the following notation: Let $\lambda$ be a diagram, and $\square$ a box in $\lambda,$ and define the upper hook length and lower hook length as

\begin{align} h_\lambda(\square) & \coloneqq (a \cdot \arm_\lambda(\square) + \leg_\lambda(\square) + 1) \\ h'_\lambda(\square) & \coloneqq (a \cdot \arm_\lambda(\square) + \leg_\lambda(\square) + a). \end{align}

These are denoted $h^*_\lambda(s)$ and $h_*^\lambda(s),$ respectively, in [Sta89].

Define the two $a$-deformations of the product of hook values in the diagram $\lambda$:

\[ H_\lambda = \prod_{\square \in \lambda} h_\lambda(\square), \quad H'_\lambda = \prod_{\square \in \lambda} h'_\lambda(\square). \]

The relationship between $\jackJ_\lambda$ and $\jackP_\lambda$ is then given by $\jackJ_\lambda(x;a) = H_\lambda \jackP_\lambda(x;a).$ Furthermore, we have that

\[ \langle \jackJ_\lambda, \jackJ_\lambda \rangle_a = H_\lambda H'_\lambda \quad \text{and} \quad \langle \jackJ_\mu \jackJ_\nu, \jackJ_\lambda \rangle_a = H'_\lambda H_\mu H_\nu \langle \jackP_\mu \jackP_\nu, \jackP_\lambda \rangle_a . \]
Example.

For example,

\[ \jackJ_{31}(x;a) = (2a^2 + 4a + 2)\monomial_{31}+(6a + 10)\monomial_{211}+(4a + 4)\monomial_{22}+ 24\monomial_{1111}. \]

Specializations

We have that

\[ \jackP_\lambda(x;1) =\schurS_\lambda(x),\quad \jackP_{\lambda'}(x;0)=\elementaryE_{\lambda}(x) \text{ and } \jackP_{\lambda}(x;\infty) = \monomial_{\lambda}(x). \]

Similarly,

\[ \jackJ_\lambda(x;1) = H_\lambda \schurS_\lambda(x), \quad \jackJ_{\lambda'}(x;0) = \lambda! \elementaryE_{\lambda}(x) \text{ and } \jackJ_{\lambda}(x;\infty) = n! \monomial_{\lambda}(x). \]

We also have that $\jackJ_\lambda(\xvec;2) = \zonal_\lambda(\xvec),$ the Zonal symmetric functions.

Cauchy identity

Recall that $H_\lambda H'_\lambda = \langle \jackJ_\lambda, \jackJ_\lambda \rangle_a.$ The Cauchy identity for Jack polynomials states that

\[ \sum_{\lambda} \frac{\jackJ_\lambda(x;a) \jackJ_\lambda(y;a)}{H_\lambda H'_\lambda} = \prod_{i,j} (1-x_i y_j)^{-1/a}. \]

Calogero–Sutherland and Laplace–Beltrami operators

The Jack polynomials $\jackJ_\lambda$ are eigenpolynomials for the Calogero–Sutherland operator

\[ \mathcal{H} \coloneqq \frac{\alpha}{2} \sum_{i=1}^n \left(x_i\frac{\partial}{\partial x_i}\right)^2 + \frac{1}{2} \sum_{i\lt j} \left( \frac{x_i+x_j}{ x_i - x_j } \right)\left( x_i \frac{\partial}{\partial x_i} - x_j \frac{\partial}{\partial x_j} \right), \]

so that

\[ \mathcal{H} \jackJ_\lambda = \sum_{i=1}^n \left( \frac{\alpha}{2} \lambda_i^2 + \frac{n+1-2i}{2} \right) \jackJ_\lambda. \]

See [Sut71] for the physics background of $\mathcal{H}.$

The are also eigenpolynomials for the (quasi) Laplace–Beltrami operator,

\[ \frac{\alpha}{2} \sum_{i=1}^n \left(x_i\frac{\partial}{\partial x_i}\right)^2 + \frac{1}{2} \sum_{i\neq j} \left( \frac{x_i^2}{ x_i - x_j } \right) \frac{\partial}{\partial x_i}. \]

See [LLM99Rob00] for background.

Knop–Sahi combinatorial formula

In [KS97], the following formula for the monomial expansion of the integral form Jack polynomials was found:

\[ \jackJ_\lambda (x;a) = \sum_{T \in \mathrm{NAF}(\lambda)} d_T(a) x^T \]

where $\mathrm{NAF}(\lambda)$ is the set of non-attacking fillings of the diagram $\lambda.$ These are fillings of $\lambda$ with natural numbers such that for all boxes $(i,j),$ we have

  • $T(i,j) \neq T(i',j)$ whenever $i \neq i',$
  • $T(i,j) \neq T(i',j+1)$ whenever $i \gt i'.$

The quantity $d_T(a)$ is defined as

\[ d_T(a) = \prod_{s \in crit(T)} [a( \arm_\lambda(s) +1 ) + ( \leg_\lambda(s) +1 ) ] \]

and $crit(T)$ is the set of boxes $(i,j)$ with $j>1$ such that $T(i,j) = T(i,j-1).$ The Knop–Sahi formula follows from the more general combinatorial formula for Macdonald polynomials in [HHL05].

In a recent paper [NSS21], the Knop–Sahi formula is generalized to the non-symmetric, interpolation setting.

Pieri rule

R. Stanley provides a Pieri rule for Jack polynomials.

Theorem (See [Thm. 6.1, Sta89]).

Let $\mu \subseteq \lambda$ and let $\lambda/\mu$ be a horizontal $r$-strip. Then

\begin{equation*} \langle \jackJ_{(r)}\jackJ_{(\mu)}, \jackJ_\lambda \rangle = \left( \prod_{\square \in \mu} A_{\lambda \mu}(\square) \right) \left( \prod_{\square \in (r)} h_{(r)}(\square) \right) \left( \prod_{\square \in \lambda} B_{\lambda \mu}(\square) \right) \end{equation*}

where

\begin{align} A_{\lambda \mu}(\square) &= \begin{cases} h'_\mu(\square) \text{ if $\lambda/\mu$ does does not intersect the column of $\square$} \\ h_\mu(\square) \text{ otherwise,} \\ \end{cases} \\ B_{\lambda \mu}(\square) &= \begin{cases} h_\lambda(\square) \text{ if $\lambda/\mu$ does does not intersect the column of $\square$} \\ h'_\lambda(\square) \text{ otherwise.} \\ \end{cases} \end{align}

The middle product is simply $r!a^r.$

A dual Pieri rule, for computing $\langle \jackJ_{(1^r)}\jackJ_{(\mu)}, \jackJ_\lambda \rangle$ is given in [Thm. 6.1, KS96]. It is given for the Jack $P$ functions, and is stated as follows. Let $X(\lambda/\mu)$ be the set of boxes $(i,j)$ in $\mu,$ such that $\mu_i=\lambda_i$ and $\mu'_j \lt \lambda'_j.$ Then

\[ \langle \jackP_{(1^r)}\jackP_{(\mu)}, \jackP_\lambda \rangle = \prod_{\square \in X(\lambda/\mu)} \frac{ h'_\lambda(\square) h_\mu(\square)}{h_\lambda(\square) h'_\mu(\square)}. \]

There is a nice symmetry for the Littlewood–Richardson coefficients, where conjugation of all three shapes correspond to swapping upper and lower hooks, see Eq. 2.3.1 in https://rucore.libraries.rutgers.edu/rutgers-lib/44186/PDF/1/play/.

Jack in power-sum basis

In [HW17], a formula for $\jackJ_\lambda (x;a)$ in terms of power-sum symmetric functions is given. It is in general not cancellation free.

A formula for the Schur expansion is also given, but it is fairly complicated and not cancellation free in general.

Jack Littlewood–Richardson conjecture

Conjecture (See [Sta89]).

R. Stanley conjecture that the coefficients $g^\lambda_{\mu\nu}(a) \coloneqq H'_\lambda H_\mu H_\nu \langle \jackP_\mu \jackP_\nu, \jackP_\lambda \rangle_a = \langle \jackJ_\mu \jackJ_\nu, \jackJ_\lambda \rangle_a$ are polynomials in $a$ with non-negative integer coefficients.

Polynomiality of $g^\lambda_{\mu\nu}(a)$ has been proved [KS97], so only the non-negativity result remains open. This conjecture has a generalization for shifted Jack polynomials.

Lemma.

The coefficients $g^\lambda_{\mu\nu}(a)$ can be computed via

\[ \jackJ_\mu \jackJ_\nu = \sum_{\lambda} \frac{g^\lambda_{\mu\nu}(a)}{H_\lambda H'_\lambda} \jackJ_\lambda. \]
Proof.

Since $\jackJ_\mu = H_\mu\jackP_\mu,$ we have that

\begin{align} H_\mu \jackP_\mu \cdot H_\nu \jackP_\nu &= \sum_{\lambda} \frac{g^\lambda_{\mu\nu}(a)}{H'_\lambda \cdot H_\lambda} H_\lambda \jackP_\lambda, \end{align}

and by rearranging the factors,

\begin{align} \jackP_\mu \jackP_\nu &= \sum_{\lambda} \frac{g^\lambda_{\mu\nu}(a)}{H_\mu \cdot H_\nu \cdot H'_\lambda} \jackP_\lambda. \end{align}

This implies that $g^\lambda_{\mu\nu}(a) = H'_\lambda H_\mu H_\nu \langle \jackP_\mu \jackP_\nu, \jackP_\lambda \rangle_a.$

Skew Jack polynomials

There is no Knop–Sahi analog known for skew Jack polynomials, and there is no combinatorial formula for the monomial expansion of skew Jack polynomials (even though the coefficients are conjectured to be in $\setN[a]$). Some observations are proved in [BG21], and it is clear that this conjecture is very much related to the positivity of $g^\lambda_{\mu\nu}(a).$

Hanlon's conjecture

Conjecture (See [Han88]).

Hanlon conjectured that there is some weight function $w(\sigma,\tau),$ such that

\begin{equation*} \jackJ_\lambda(x;a) = \sum_{\substack{\sigma \in RS(\lambda) \\ \tau \in CS(\lambda)}} \sign(\sigma) a^{w(\sigma,\tau)} \powerSum_{\text{type}(\sigma\tau)}(x) \end{equation*}

where $RS(\lambda)$ and $CS(\lambda)$ is the row- and column-stabilizers of a fixed standard Young tableau of shape $\lambda.$

Schur expansion conjecture

Conjecture (See [AHW18]).

We conjecture that the coefficients $c_k(\mu,\lambda)$ the expansion

\begin{equation*} \langle a^{|\lambda|}\jackJ_\mu(x;1/a) , \schurS_\lambda(x) \rangle = \sum_{k=0}^{n-1} c_k(\mu,\lambda) \binom{a+k}{n} \end{equation*}

are non-negative integers. Furthermore, in the expansion

\begin{equation*} \langle a^{|\lambda|}\jackJ_\mu(x;1/a) , \schurS_\lambda(x) \rangle = \sum_{k=1}^{n-1} b_{n-k}(\mu,\lambda) \binom{a}{k} k! \end{equation*}

we believe that $\sum_{k=0}^n b_{n-k}(\mu,\lambda) z^k \in \setN[z]$ and that all roots of such polynomials are real.

This conjecture has a rich interplay with rook polynomials, and the relationship between the $c_k(\mu,\lambda)$ and $b_{n-k}(\mu,\lambda)$ are similar to that of rook hit polynomials and rook polynomials.

Gessel expansion conjecture

Conjecture (See [AHW18]).

We conjecture that there is a statistic $\sigma,$ such that

\begin{equation*} a^{|\lambda|}\jackJ_\lambda(x;1/a) = \sum_{\pi,\tau \in \symS_n} \binom{a+n-1-\des(\pi)}{n} \gessel_{\sigma(\mu,\pi,\tau)}(x). \end{equation*}

References