The symmetric functions catalog

An overview of symmetric functions and related topics

2020-09-07

Hall–Littlewood P

The Hall–Littlewood $P$ polynomals interpolates between the Schur polynomals and the monomial symmetric functions. They were introduced by P. Hall (implicitly) and later by D. E. Littlewood in [Lit61].

Definition

For a partition $\lambda,$ with $\alpha_i$ parts equal to $i,$ we define the Hall–Littlewood functions as

\begin{equation*} \hallLittlewoodP_\lambda(x_1,\dotsc,x_n;t) = \frac{1}{\prod_{i\geq 0} [\alpha_i]_t! }\sum_{\sigma \in \symS_n} \sigma\left( \xvec^\lambda \frac{ \prod_{i \lt j} 1-tx_j/x_i }{\prod_{i \lt j} 1-x_j/x_i } \right). \end{equation*}

Compare with the Weyl formula for Schur polynomials. Note that

\[ \hallLittlewoodP_\lambda(\xvec;1) = \monomial_\lambda(\xvec) \text{ and } \hallLittlewoodP_\lambda(\xvec;0) = \schurS_\lambda(\xvec). \]

Furthermore, $\hallLittlewoodP_\lambda(\xvec;-1)$ is equal to the Schur-$P$ function.

The Hall–Littlewood polynomials are orthogonal with respect to the inner product $\langle - , - \rangle_t$ defined via

\[ \langle \powerSum_\lambda , \powerSum_\mu \rangle_t = \delta_{\lambda\mu} z_{\lambda} \prod_{i=1}^{\length(\lambda)} \frac{1}{1-t^{\lambda_i} }, \]

see [III. §4, Mac95]. This is a $t$-deformation of the usual Hall inner product, which is recovered at $t=0.$

The (skew) Hall–Littlewood $\hallLittlewoodP$ functions can be computed via a branching rule as follows. First let the one-variable specialization be $\hallLittlewoodP_{\lambda/\mu}(z;t) = z^{|\lambda/\mu|} \psi_{\lambda/\mu}(t),$ where

\[ \psi_{\lambda/\mu}(t) = \prod_{\substack{ i \geq 1 \\ m_i(\mu) = m_i(\lambda)+1 }} \left( 1-t^{m_i(\mu)} \right). \]

Then the skew Hall–Littlewood $\hallLittlewoodP$ polynomial is given by

\[ \hallLittlewoodP_{\lambda/\mu}(x_1,\dotsc,x_n;t) = \sum_{ \mu = \nu^{0} \prec \dotsb \prec \nu^n = \lambda} \prod_{i=1}^n \psi_{\nu^{i}/\nu^{i-1}}(t) x_i^{|\nu^{i}/\nu^{i-1}|}. \]

Here, we write $\mu \prec \lambda$ if $\lambda/\mu$ is a skew shape.

We have that the Hall–Littlewood $\hallLittlewoodP$ polynomials is the dual basis to the Hall–Littlewood $\hallLittlewoodQ$ basis, $\langle \hallLittlewoodQ_\lambda, \hallLittlewoodP_\lambda \rangle_t = \delta_{\lambda\mu}.$

Cauchy identity

We have the Cauchy identity (see [Chapter III, Eq. (4.4), Mac95]),

\[ \prod_{i=1}^n \prod_{j=1}^n \left( \frac{1-tx_i y_j}{1-x_iy_j } \right) = \sum_{\mu} \hallLittlewoodP_\mu(x_1,\dotsc,x_n;t) \hallLittlewoodQ_\mu(y_1,\dotsc,y_n;t). \]

Pieri Rule

In [KL12], the authors prove a skew Pieri rule for the Hall–Littlewood $\hallLittlewoodP$ functions. Below is the simpler non-skew version.

Theorem (See [Thm. 1, KL12]).

Let $\lambda$ be a partition and $r \geq 0.$ Then

\[ \hallLittlewoodP_\mu(\xvec;t) \elementaryE_r(\xvec) = \sum_{\substack{ \lambda \\ |\lambda/\mu| = r }} sk_{\lambda/\mu}(t) \hallLittlewoodP_\lambda(\xvec;t) \]

where

\[ sk_{\lambda/\mu}(t) = t^{\sum_{j} \binom{\lambda'_j - \mu'_j}{2} } \prod_{i\geq i} \qbinom{ \lambda'_j - \mu'_{j+1}}{ m_j(\mu) }_t. \]

Kostka–Foulkes polynomials

The polynomials $K_{\lambda\mu}(t)$ in the expansion

\[ \schurS_\lambda(\xvec) = \sum_\mu K_{\lambda\mu}(t) \hallLittlewoodP_\mu(\xvec) \]

are called Kostka–Foulkes polynomials and are certain $t$-analogues of the Kostka coefficients. It was shown by Lascoux and Schutzenberger [LS78] that

\[ K_{\lambda\mu}(t) = \sum_{T \in \mathrm{SSYT}(\lambda,\mu)} t^{\charge(T)}, \]

where the sum is over semi-standard Young tableaux and charge is a certain statistic on SSYTs. There is an alternative way to define Kostka–Foulkes polynomials using rigged configurations, introduced by Kirillov and Reshetikhin [KR88]. See the page on Kostka–Foulkes polynomials for more information.

Products

There are various different products that lead to $t$-deformations of known structure constants.

Hall polynomials

The Hall polynomials, $c^{\nu}_{\lambda\mu}(t) \in \setZ[t]$ deform the Littlewood–Richardson coefficients. They are defined via the expansion

\[ \hallLittlewoodP_\lambda(\xvec) \hallLittlewoodP_\mu(\xvec) = \sum_\nu c^{\nu}_{\lambda\mu}(t) \hallLittlewoodP_\nu(\xvec), \]

and at $t=0,$ the Littlewood–Richardson coefficients for Schur polynomials are recovered. Note that the polynomials $c^{\nu}_{\lambda\mu}(t)$ may have negative coefficients.

A combinatorial model for $c^{\nu}_{\lambda\mu}(t)$ can be recovered from the more general theory in [Yip12]. An earlier proof appear in [Sch06].

In [Zin19], a honeycomb model for the Hall polynomials is described. This extends the honeycomb model by Knutson–Tao for the Littlewood–Richardson coefficients.

The Hall polynomials have significance in the study of abelian $p$-groups. Let $p$ be a prime number. An abelian $p$-group of type $\lambda$ is isomorphic to

\[ G = \setZ/p^{\lambda_1} \oplus \setZ/p^{\lambda_2} \oplus \dotsb \oplus \setZ/p^{\lambda_\ell}. \]

A subgroup $H \subseteq G$ has cotype $\nu$ if $G/H$ has type $\nu.$

Theorem.

The number of subgroups of type $\mu$ and cotype $\nu$ in a finite abelian $p$-group of type $\lambda$ is given by $c^{\lambda}_{\mu\nu}(p).$

For fixed $\lambda,$ the Hall polynomial $c^{\lambda}_{\mu\nu}(p)$ has non-negative coefficients for all $\mu$ and $\nu$ if and only if no two parts of $\lambda$ differ by more than one.

The parameters for which all coefficients are positive is determined exactly in [BH93]. This result later inspired a proof of the following theorem.

Theorem (See [Mal96]).

The shifted polynomial $c^{\lambda}_{\mu\nu}(p+1)$ lie in $\setN[p].$

Other deformations

In [Thm. 4, WZ18], the authors give a (signed) combinatorial formula for $\overline{K}^{\nu}_{\lambda\mu}(t)$ in the product

\[ \hallLittlewoodP_\lambda(\xvec;t) \schurS_\mu(\xvec) = \sum_{\nu} \overline{K}^{\nu}_{\lambda\mu}(t) \schurS_\nu(\xvec). \]

Hall–Littlewood Q

The Hall–Littlewood $Q$ polynomials can be defined as

\begin{equation*} \hallLittlewoodQ_\lambda(x_1,\dotsc,x_n;t) = (1-t)^{\length(\lambda)}[n-\length(\lambda)]_t! \sum_{\sigma \in \symS_n} \sigma\left( \xvec^\lambda \frac{ \prod_{i \lt j} 1-tx_j/x_i }{\prod_{i \lt j} 1-x_j/x_i } \right). \end{equation*}

Thus, $\hallLittlewoodQ_\lambda(\xvec;t)$ and $\hallLittlewoodP_\lambda(\xvec;t)$ differ only by a factor in $\setZ[t].$

Theorem (See [Eq. (17), DLT94] ).

Littlewood proved that

\[ \hallLittlewoodQ_\lambda(\xvec;q) = \prod_{i \lt j} (1-qR_{ij})^{-1}\schurS_\lambda[(1-q)X] \]

where we use raising operators and plethystic notation.

Transformed Hall–Littlewood

The transformed Hall–Littlewood polynomials are defined via the Kostka–Foulkes polynomials as

\[ \hallLittlewoodT_\mu(\xvec;q) \coloneqq \sum_{\lambda} K_{\lambda\mu}(q) \schurS_\lambda(\xvec). \]

These are sometimes also denoted $Q'_\mu(\xvec;q),$ some references are [DLT94] and [TZ03]. It is shown that

\[ \hallLittlewoodT_\mu(\xvec;q) = \prod_{i \lt j} \frac{1-R_{ij}}{1-q R_{ij}} \completeH_{\mu}(\xvec) = \prod_{i \lt j} (1-q R_{ij})^{-1} \schurS_{\mu}(\xvec) \]

where we use the same notation as in the raising operator formula for Schur polynomials. In particular, from this deinition it is clear that $\hallLittlewoodT_\mu(\xvec;0) = \schurS_\mu(\xvec).$

The relationship between the classical and transformed Hall–Littlewood functions is given by the plethystic relationship

\[ \hallLittlewoodQ_\mu(\xvec;q) = \hallLittlewoodT_\mu[(1-q)\xvec;q]. \]

In [TZ03], a $q$-deformation of Schur's Q-functions is introduced in a manner analogously to the construction above.

Example.

The transformed Hall–Littlewood polynomials indexed by partitions of size four are given as

\begin{align*} \hallLittlewoodT_{4}(\xvec;q) &= \schurS_{4} \\ \hallLittlewoodT_{31}(\xvec;q) &= q \schurS_{4} + \schurS_{31} \\ \hallLittlewoodT_{22}(\xvec;q) &=q^2 \schurS_{4}+\schurS_{22} + q\schurS_{31} \\ \hallLittlewoodT_{211}(\xvec;q) &=q^3 \schurS_{4} + q \schurS_{22} + (q+q^2)\schurS_{31}+ \schurS_{211} \\ \hallLittlewoodT_{1111}(\xvec;q) &=q^6 \schurS_{4} + (q^2+q^4)\schurS_{22} + (q^3+q^4+q^5)\schurS_{31}+(q+q^2+q^3) \schurS_{211} + \schurS_{1111} \end{align*}

The coefficients are the Kostka–Foulkes polynomials.

The transformed Hall–Littlewood polynomials are special cases of the Catalan symmetric functions. Using these, one can define compositional Hall–Littlewood polynomials, $\hallLittlewoodT_{\gamma}(\xvec;q)$ where $\gamma$ is a composition.

We also have the following relation with non-symmetric Macdonald polynomials:

\[ \omega\hallLittlewoodT_{\lambda'}(x_1,\dotsc,x_n;q) = \macdonaldE_{\rev(\lambda)}(x_1,\dotsc,x_n;q,0). \]

A consequence of this relation is the following identity involving LLT polynomials:

\[ q^{\sum_{i\geq 2} \binom{\lambda_i}{2} } \omega \hallLittlewoodH_\lambda(\xvec;t) = \LLT_{\nuvec}(\xvec;q) \]

where $\nuvec = (1^{\lambda_1}, 1^{\lambda_2},\dotsc,1^{\lambda_\ell}).$

Theorem (See proof of [Cor. 40, Ale20]).

The function $\hallLittlewoodT_\mu(\xvec;q+1)$ expands in the complete homogeneous symmetric basis with coefficients in $\setN[q].$

An explicit combinatorial expansion is given in [AS20]:

\[ \hallLittlewoodT_{\mu'}(\xvec;q+1) = (q+1)^{-\sum_{i\geq 2} \binom{\mu_i}{2} } \sum_{\theta \in \mathcal{O}(P_\mu)} q^{\asc(\theta)} \completeH_{\lambda(\theta)}(\xvec). \]

Here, $\lambda(\theta)$ is a certain statistic on orientations of a graph, indexed by a Schroder path $P_\mu.$

Modified Hall–Littlewood polynomials

The modified Hall–Littlewood polynomials are related to the transformed Hall–Littlewood polynomials via

\[ \hallLittlewoodH_\lambda(\xvec;t) = t^{\partitionN(\lambda)} \hallLittlewoodT_\lambda(\xvec;t^{-1}). \]

The modified Hall–Littlewood polynomials are more combinatorial in nature. They are Schur-positive and are in fact the Frobenius image of a certain graded vector space.

The modified Hall–Littlewood polynomials are specializations of the the modified Macdonald polynomials.

Modified Kostka–Foulkes polynomials

The modified Kostka–Foulkes polynomials are defined as

\[ \tilde{K}_{\lambda\mu}(t) = t^{\partitionN(\mu)}K_{\lambda\mu}(1/t) = \sum_{T \in \mathrm{SSYT}(\lambda,\mu)} t^{\cocharge(T)}, \]

where the sum is over semi-standard Young tableaux and cocharge is defined via the charge statistic.

We then have that

\[ \hallLittlewoodH_\mu(\xvec;t) = \sum_{\lambda} \tilde{K}_{\lambda\mu}(t) \schurS_{\lambda}(\xvec). \]

In other root systems

Hall–Littlewood polynomials can be define in general for symmetrizable Kac–Moody algebras, see http://www.kurims.kyoto-u.ac.jp/EMIS/journals/SLC/wpapers/s58viswa.pdf

References