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12.2 Hall-Littlewood Polynomials

7 min read•july 30, 2024

Hall-Littlewood_Polynomials_0### are a family of symmetric functions that bridge Schur functions and monomial symmetric functions. They're indexed by partitions and depend on a parameter t, forming a basis for symmetric functions over rational functions in t.

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These polynomials have applications in representation theory, algebraic geometry, and combinatorics. They're related to , specializing to them when q=0, and connect to , which count Young tableaux with a certain charge statistic.

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Hall-Littlewood polynomials

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Definition and properties

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  • Hall- polynomials, denoted as Pλ([x](https://www.fiveableKeyTerm:x);t)P_\lambda([x](https://www.fiveableKeyTerm:x);t), are a family of symmetric polynomials indexed by partitions λ\lambda and depend on a parameter tt
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  • Form a basis for the ring of symmetric functions over the field of rational functions in tt
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  • Satisfy a triangularity property with respect to the dominance order on partitions
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- For partitions $\lambda$ and $\mu$, $P_\lambda(x;t)$ can be expressed as a linear combination of $P_\mu(x;t)$ with coefficients in $\mathbb{Z}[t]$, where the sum is over $\mu \leq \lambda$ in the dominance order
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  • Orthogonal with respect to a certain inner product on the ring of symmetric functions
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- The inner product is defined by $\langle p_\lambda, p_\mu \rangle = z_\lambda \delta_{\lambda\mu} \prod_{i \geq 1} (1-t^{m_i(\lambda)})$, where $p_\lambda$ are the power sum symmetric functions and $m_i(\lambda)$ is the number of parts of $\lambda$ equal to $i$
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  • Can be defined using a modified version of the Kostka numbers, known as the Kostka-Foulkes polynomials
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- $P_\lambda(x;t) = \sum_{\mu} K_{\lambda\mu}(t) m_\mu(x)$, where $m_\mu(x)$ are the monomial symmetric functions and $K_{\lambda\mu}(t)$ are the Kostka-Foulkes polynomials
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Specializations and identities

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  • Specialize to several other important families of symmetric functions
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- Schur functions: $P_\lambda(x;0) = s_\lambda(x)$
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- Monomial symmetric functions: $P_\lambda(x;1) = m_\lambda(x)$
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  • Satisfy a , which relates them to the elementary and complete homogeneous symmetric functions
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- $\sum_{\lambda} P_\lambda(x;t) Q_\lambda([y](https://www.fiveableKeyTerm:y);t) = \prod_{i,j} (1-x_i y_j)^{-1}$, where $Q_\lambda(y;t)$ are the dual Hall-Littlewood polynomials defined by $Q_\lambda(y;t) = b_\lambda(t) P_\lambda(y;t^{-1})$ with $b_\lambda(t) = \prod_{i \geq 1} (t^{m_i(\lambda)}-1) / (t-1)$
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Computing Hall-Littlewood polynomials

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Combinatorial formula

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  • Can be computed using a combinatorial formula involving tableaux filled with positive integers
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- Entries in each row of the tableau must be weakly increasing, and the entries in each column must be strictly increasing
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- Weight of a tableau is a product of factors involving the parameter $t$ and the number of entries in each row
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- $P_\lambda(x;t) = \sum_{T} x^T \prod_{i \geq 1} (t^{n_i(T)}-1) / (t-1)$, where the sum is over all semistandard Young tableaux $T$ of shape $\lambda$, $x^T$ is the monomial obtained by taking the product of $x_i$ for each entry $i$ in $T$, and $n_i(T)$ is the number of entries equal to $i$ in $T$
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Determinantal and recursive formulas

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  • Can be computed using a determinantal formula involving the elementary symmetric functions
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- $P_\lambda(x;t) = \det(e_{\lambda_i'-j+1}(x;t))_{1 \leq i,j \leq \ell(\lambda')}$, where $\lambda'$ is the conjugate partition of $\lambda$ and $e_k(x;t)$ are the modified elementary symmetric functions defined by $\sum_{k \geq 0} e_k(x;t) z^k = \prod_{i \geq 1} (1 + x_i z) / (1 - t x_i z)$
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  • Can be computed recursively using a branching rule that relates Pλ(x;t)P_\lambda(x;t) to Hall-Littlewood polynomials indexed by partitions obtained by removing a part from λ\lambda
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- $P_\lambda(x;t) = \sum_{i: \lambda_i > \lambda_{i+1}} (1 - t^{m_i(\lambda)}) x_i P_{\lambda^{(i)}}(x;t)$, where $\lambda^{(i)}$ is the partition obtained from $\lambda$ by removing a part of size $i$
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Generating function and vertex operator formulas

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  • Can be computed using a generating function that involves the complete homogeneous symmetric functions and a plethystic substitution
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- $\sum_{\lambda} P_\lambda(x;t) z^\lambda = \exp\left(\sum_{k \geq 1} \frac{1}{k} \frac{1-t^k}{1-t} p_k(x) p_k(z)\right)$, where $p_k(x)$ and $p_k(z)$ are the power sum symmetric functions in the variables $x$ and $z$, respectively
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  • Can be computed using a vertex operator formula that involves certain infinite matrices
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- $P_\lambda(x;t) = \langle v_\lambda, \Gamma_+(x) v_0 \rangle$, where $v_\lambda$ and $v_0$ are basis vectors in a certain infinite-dimensional vector space, $\Gamma_+(x)$ is a vertex operator defined using infinite matrices, and $\langle \cdot, \cdot \rangle$ is a certain bilinear form
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Hall-Littlewood polynomials and symmetric functions

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Relationship to other bases

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  • Are a basis for the ring of symmetric functions, which is a subring of the ring of polynomials in infinitely many variables
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  • Schur functions, which are another important basis for the ring of symmetric functions, can be obtained from the Hall-Littlewood polynomials by setting t=0t=0
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- $s_\lambda(x) = P_\lambda(x;0)$
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Kostka-Foulkes polynomials and Young tableaux

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  • Kostka-Foulkes polynomials, which are the coefficients in the expansion of the Schur functions in terms of the Hall-Littlewood polynomials, have a combinatorial interpretation in terms of Young tableaux
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- The Kostka-Foulkes polynomial $K_{\lambda\mu}(t)$ is the generating function for the set of semistandard Young tableaux of shape $\lambda$ and content $\mu$, where the power of $t$ counts the charge of the tableau
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- $s_\lambda(x) = \sum_{\mu} K_{\lambda\mu}(t) P_\mu(x;t)$
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  • Can be used to define a tt-analog of the Kostka numbers, which count the number of semistandard Young tableaux of a given shape and content
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- $K_{\lambda\mu}(t) = \sum_{T} t^{\mathrm{charge}(T)}$, where the sum is over all semistandard Young tableaux $T$ of shape $\lambda$ and content $\mu$
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Relationship to Macdonald polynomials

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  • Are related to the Macdonald polynomials, which are a two-parameter generalization of the Schur functions and the Hall-Littlewood polynomials
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- Macdonald polynomials $P_\lambda(x;q,t)$ specialize to Hall-Littlewood polynomials when $q=0$: $P_\lambda(x;0,t) = P_\lambda(x;t)$
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- Macdonald polynomials specialize to Schur functions when $q=t=0$: $P_\lambda(x;0,0) = s_\lambda(x)$
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Applications of Hall-Littlewood polynomials

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Representation theory of symmetric and general linear groups

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  • Appear in the representation theory of the symmetric group and the general linear group
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- Can be used to construct certain modules for the symmetric group, known as the permutation modules
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    - The permutation module $M^\lambda$ corresponding to a partition $\lambda$ has a basis indexed by the set of Young tableaux of shape $\lambda$
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- Decomposition of the permutation modules into irreducible representations is governed by the Kostka-Foulkes polynomials
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    - $M^\lambda \cong \bigoplus_{\mu} K_{\lambda\mu}(1) S^\mu$, where $S^\mu$ is the irreducible representation of the symmetric group corresponding to the partition $\mu$
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Affine Lie algebras and Demazure characters

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  • Are related to the Demazure characters of certain affine Lie algebras
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- Demazure characters are certain truncations of the character of an irreducible highest weight representation of an affine Lie algebra
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- Hall-Littlewood polynomials appear as special cases of Demazure characters for the affine Lie algebra $\widehat{\mathfrak{sl}}_n$
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Cohomology of Grassmannian and flag varieties

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  • Can be used to study the cohomology of the Grassmannian variety and the flag variety
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- The Grassmannian variety $\mathrm{Gr}(k,n)$ is the set of $k$-dimensional subspaces of an $n$-dimensional vector space
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- The cohomology ring of $\mathrm{Gr}(k,n)$ has a basis given by the Schubert classes, which are indexed by partitions $\lambda$ contained in a $k \times (n-k)$ rectangle
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- The Hall-Littlewood polynomials appear as representatives for the Schubert classes in the cohomology ring of $\mathrm{Gr}(k,n)$
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Hilbert scheme of points and Heisenberg algebra

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  • Appear in the study of the Hilbert scheme of points on a surface and its connection to the representation theory of the Heisenberg algebra
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- The Hilbert scheme $\mathrm{Hilb}^n(S)$ of $n$ points on a surface $S$ is a smooth algebraic variety that parametrizes zero-dimensional subschemes of $S$ of length $n$
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- The cohomology ring of $\mathrm{Hilb}^n(\mathbb{C}^2)$ has a basis given by the Schur functions $s_\lambda(x)$, where $\lambda$ is a partition of $n$
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- The Hall-Littlewood polynomials appear in the study of the equivariant cohomology of $\mathrm{Hilb}^n(\mathbb{C}^2)$ with respect to the natural action of the torus $(\mathbb{C}^*)^2$
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Kazhdan-Lusztig polynomials and Hecke algebras

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  • Are related to the Kazhdan-Lusztig polynomials, which are important in the representation theory of Hecke algebras and quantum groups
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- The Kazhdan-Lusztig polynomials $P_{y,w}(q)$ are polynomials in $q$ indexed by pairs of elements $y,w$ in a Coxeter group $W$
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- The Kazhdan-Lusztig polynomials for the symmetric group $S_n$ are closely related to the Hall-Littlewood polynomials
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    - For permutations $y,w \in S_n$, $P_{y,w}(q) = K_{\lambda(y),\lambda(w)}(q)$, where $\lambda(y)$ and $\lambda(w)$ are the partitions obtained by sorting the code of the permutations $y$ and $w$, respectively
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- The Kazhdan-Lusztig polynomials for the affine Weyl group of type $\widetilde{A}_{n-1}$ are related to the Hall-Littlewood polynomials for the affine root system of type $A_{n-1}^{(1)}$
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