The symmetric functions catalog

An overview of symmetric functions and related topics

2024-02-22

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].

Cauchy identity

The (dual) Cauchy identity for Jack $P$ polynomials states that

\[ \sum_{\lambda} \jackP_\lambda(x;a) \jackP_\lambda(y;1/a) = \prod_{i,j} (1+x_i y_j), \]

see [Eq. (2.6), Mac92]. There is also a generalization of the Cauchy identity for the Jack J polynomials.

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) + a) \\ h_\lambda(\square) & \coloneqq (a \cdot \arm_\lambda(\square) + \leg_\lambda(\square) + 1). \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 coefficients

Conjecture (See [Sta89]).

R. Stanley conjecture that the coefficients $g^\lambda_{\mu\nu}(a) \coloneqq \langle \jackJ_\mu \jackJ_\nu, \jackJ_\lambda \rangle_a = H_\lambda H_\mu H_\nu \langle \jackP_\mu \jackP_\nu, \jackP_\lambda \rangle_a$ 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. The case when the indexing partitions has at most three parts is proved in [Naq], and another case involving rectangular shapes is considered in [CJ].

This conjecture has a generalization for shifted Jack polynomials.

Lemma.

Let us define the Jack Littlewood–Richardson coefficients $c^\lambda_{\mu\nu}(a)$ via

\[ \jackP_\mu \jackP_\nu = \sum_{\lambda} c^\lambda_{\mu\nu}(a) \jackP_\lambda. \]

Then $g^\lambda_{\mu\nu}(a) = H'_\lambda H_\mu H_\nu c^\lambda_{\mu\nu}(a).$

Proof.

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

\begin{align} \frac{\jackJ_\mu}{H_\mu} \frac{\jackJ_\nu}{H_\nu} = \sum_{\lambda} c^\lambda_{\mu\nu}(a) \frac{\jackJ_\lambda}{H_\lambda} \end{align}

and by rearranging the factors,

\[ \jackJ_\mu \jackJ_\nu = \sum_{\lambda} \frac{H_\mu \cdot H_\nu \cdot c^\lambda_{\mu\nu}(a)}{H_\lambda} \jackJ_\lambda. \]

We now apply $\langle \cdot , \jackJ_\lambda \rangle_a$ on both sides and get

\[ \langle \jackJ_\mu \jackJ_\nu, \jackJ_\lambda \rangle_a = \frac{H_\mu \cdot H_\nu \cdot c^\lambda_{\mu\nu}(a)}{H_\lambda} \langle \jackJ_\lambda , \jackJ_\lambda \rangle_a. \]

This implies that

\[ g^\lambda_{\mu\nu}(a) = H'_\lambda H_\mu H_\nu \cdot c^\lambda_{\mu\nu}(a) \]

since $\langle \jackJ_\lambda , \jackJ_\lambda \rangle_a = H_\lambda H'_\lambda.$

For some recent progress on the Jack Littlewood–Richardson coefficients, see [Mic23].

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

Expanding Jack polynomials in terms of Schur functions seem to have a strong connection with rook polynomials. This is explored in [AHW18]. We pose the following conjecture in that paper.

Conjecture (Alexandersson–Haglund–Wang, 2018).

Define the coefficients $b_{n-k}(\mu,\lambda)$ and $c_k(\mu,\lambda)$ via the expansions

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

Then $b_{n-k}(\mu,\lambda)$ and $c_k(\mu,\lambda)$ are non-negative integers. Moreover, the roots of the polynomials

\[ \sum_{k=0}^{n} b_k(\mu,\lambda) z^k \qquad \text{ and } \qquad \sum_{k=0}^{n} c_k(\mu,\lambda) z^k \]

are all 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.

Example (Table of coefficients).

Consider the expansions

\[ a^{|\lambda|}\jackJ_\mu(x;1/a) = \sum_{k=0}^{n} c_k(\mu,\lambda) \binom{a+k}{n} = \sum_{k=0}^{n} b_{n-k}(\mu,\lambda) \binom{a}{k} k! \]

and define the Jack rook polynomial $R_{\lambda,\mu}(t)$ and the Jack hit polynomial $R_{\lambda,\mu}(t)$ as

\[ R_{\lambda,\mu}(t) = \sum_{k=0}^{n} b_k(\mu,\lambda) z^k \qquad H_{\lambda,\mu}(t) = \sum_{k=0}^{n} c_k(\mu,\lambda) z^k, \]

respectively. For small $(\lambda,\mu)$ we get the following table. Missing combinations of $\lambda$ and $\mu$ means that the corresponding polynomial vanish.

$\mu$$\lambda$$\textbf{Rook}$$\textbf{Hit}$
$1$$1$$1$$1$
$2$$2$$2 z+1$$2 z$
$2$$11$$1$$2$
$11$$11$$2 z+2$$2 z+2$
$3$$3$$6 z^2+6 z+1$$6 z^2$
$3$$21$$6 z+2$$12 z$
$3$$111$$1$$6$
$21$$21$$3 z^2+7 z+2$$3 z^2+8 z+1$
$21$$111$$4 z+2$$8 z+4$
$111$$111$$6 z^2+18 z+6$$6 z^2+24 z+6$
$4$$4$$24 z^3+36 z^2+12 z+1$$24 z^3$
$4$$31$$36 z^2+24 z+3$$72 z^2$
$4$$22$$12 z^2+12 z+2$$24 z^2+24 z$
$4$$211$$12 z+3$$72 z$
$4$$1111$$1$$24$
$31$$31$$8 z^3+28 z^2+16 z+2$$8 z^3+32 z^2+8 z$
$31$$22$$12 z^2+12 z+2$$24 z^2+24 z$
$31$$211$$20 z^2+22 z+4$$40 z^2+52 z+4$
$31$$1111$$6 z+2$$36 z+12$
$22$$22$$12 z^3+48 z^2+30 z+4$$12 z^3+60 z^2+24 z$
$22$$211$$20 z^2+22 z+4$$40 z^2+52 z+4$
$22$$1111$$12 z^2+18 z+4$$24 z^2+60 z+12$
$211$$211$$8 z^3+48 z^2+38 z+6$$8 z^3+72 z^2+60 z+4$
$211$$1111$$24 z^2+30 z+6$$48 z^2+84 z+12$
$1111$$1111$$24 z^3+168 z^2+144 z+24$$24 z^3+264 z^2+264 z+24$

Data for partitions of sizes $1,2,\dotsc,9$ is available from jack-rook-hit-data.txt (106 KiB).

We conjecture that each polynomial above is real-rooted. One can show that if the Jack hit polynomials are real-rooted, then so are the Jack 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*}

Jack characters

The Jack characters are certain normalizations of the coefficients when Jack polynomials are expanded in the power-sum basis.

Let $\lambda \vdash n$ and $\mu \vdash m.$ Then the Jack character (introduced in [Las08aLas09]) is defined as

\[ \theta_{\mu}^{(\alpha)}(\lambda) \coloneqq \begin{cases} \binom{n- m + m_1(\mu)}{m_1(\mu)} \powerSum_{\mu,1^{n-m}} & \text{if } n\lt m \\ 0 & \text{if } n\lt m \end{cases} \]

where $m_1(\mu)$ denotes the number of parts equal to $1$ in $\mu.$

Lasalle conjectured that $\theta_{\mu}^{(\alpha)}(\lambda)$ satisfy a certain positivity property [Las08a]. This was recently proved by H.B. Dali and M. Dołęga [DD23]. We shall need some more definitions and background in order to state their result.

Let $\lambda = [s_1^{r_1}, s_2^{r_2}, s_k^{r_k}]$ be the multi-rectangular coordinates for $\lambda,$ where $s_1 \geq s_2 \geq \dotsb \geq s_k.$ That is, $\lambda$ consists of $k$ rectangles of size $s_i \times r_i$ stacked on top of each other. It was then proved that $\theta_{\mu}^{(\alpha)}(\mathbf{r},\mathbf{s})$ is a polynomial in $\setQ[\alpha,s_1,\dotsc,s_k,r_1,\dotsc,r_k].$ Stanley conjectured a combinatorial formula in the case $\alpha=1$ [Sta03], and this was later proved by V. Féray [Fér10].

Theorem (H.B. Dali and M. Dołęga [DD23]).

The polynomial $(-1)^{|\mu|}z_\mu \theta_{\mu}^{(\alpha)}(\mathbf{r},\mathbf{s})$ is a polynomial in the variables $\beta\coloneqq \alpha-1,$ $-s_1,\dotsc,-s_k,$ $r_1,\dotsc,r_k$ with non-negative integer coefficients.

H.B. Dali and M. Dołęga also give a combinatorial formula for $(-1)^{|\mu|}\theta_{\mu}^{(\alpha)}(\lambda)$ in terms of layered, non-oriented maps.

References