Macdonald and Hall–Littlewood Polynomials

Introduction

The Hall–Littlewood polynomials $P_\lambda(x;t)$ are a one-parameter family of symmetric functions that interpolate between two classical bases of the ring of symmetric functions: at $t = 0$ the polynomial $P_\lambda(x;0)$ is the Schur function $s_\lambda$ and at $t = 1$ it is the monomial symmetric function $m_\lambda$. The Macdonald polynomials $P_\lambda(x;q,t)$ are a two-parameter family constructed by the same scheme with the single parameter $t$ replaced by two parameters $q$ and $t$: they are the symmetric functions that are triangular with respect to the monomial basis and orthogonal with respect to a two-parameter deformation of the Hall inner product, and they specialise at $q = 0$ to the Hall–Littlewood polynomials and at $q = t$ to the Schur functions.

The article is the fifteenth of the corpus and the second of the category Symmetric Linear Algebras. It continues the account of the symmetric algebra and of its graded pieces begun in The Symmetric Algebra and Symmetric Powers, passing from the symmetric algebra of a module to the ring of symmetric functions, and it uses the deformation language of Hecke Algebras, the article immediately above it: the Hall–Littlewood polynomials are the eigenfunctions of the action of the Hecke algebra of type $A$ on the ring of symmetric functions, and the deformation parameter $t$ of the polynomials is the parameter $q$ of the Hecke algebra under the standard identification. The article develops the ring of symmetric functions and its bases, the Hall inner product, the definition of the Hall–Littlewood polynomials by symmetrisation together with the verification of the two limiting specialisations, the Kostka–Foulkes polynomials and the orthogonality, the definition of the Macdonald polynomials with their specialisations and the positivity theorem, and the Hecke-theoretic interpretation with the eigenfunction property.

Two boundaries are fixed at the outset. First, the analytic and measure-theoretic theory of the symmetric functions — the $L^2$ theory, the integration over the unitary group that occurs in the Haar-measure development of the Schur orthogonality, and the asymptotics — belongs to Part III and is not used. Second, the operators that diagonalise the Macdonald polynomials in the most general setting are the Cherednik operators of the double affine Hecke algebra; the affine Hecke algebras are objects whose analytic and geometric theory is treated in Part III, as recorded in Hecke Algebras, and only the finite Hecke algebra of type $A$ is used here. Wherever an argument would need the affine theory, the article says so. No form, no manifold, no distance and no topological object appears.

Throughout, $\Lambda$ is the ring of symmetric functions over $\mathbb{Z}$ (or over a field $k$ of characteristic $0$ after extension of scalars), $\lambda = (\lambda_1\geq\lambda_2\geq\cdots)$ is a partition with $\lvert\lambda\rvert = \sum_i\lambda_i$, $\mu\vdash n$ means $\lvert\mu\rvert = n$, $l(\lambda)$ is the number of non-zero parts, and $m_i(\lambda)$ is the multiplicity of the part $i$; the parameters are written $t$ and $q$, as in the classical notation and in Hecke Algebras, and no quadratic form occurs.

The Ring of Symmetric Functions

Definition. Let $R$ be a commutative ring with identity and let $\Lambda_R$ be the inverse limit, over the maps that send the last variable to zero, of the rings $R[x_1,\dots,x_n]^{S_n}$ of symmetric polynomials in $n$ variables. An element of $\Lambda_R$ is a symmetric function: a formal sum $\sum_{\alpha}c_\alpha x^\alpha$ with rational integer or $R$-coefficients, of bounded degree, whose monomials are the orbit sums of exponent vectors under permutation of the variables. Write $\Lambda$ for $\Lambda_{\mathbb{Z}}$.

Definition. For a partition $\lambda$ the monomial symmetric function $m_\lambda$ is the sum of the distinct monomials obtained from $x^{\lambda}$ by permuting the variables, $m_\lambda = \sum_{\alpha\in S_\lambda}x^\alpha$, where $S_\lambda$ is the orbit of $\lambda$. The elementary symmetric functions are $e_n = m_{(1^n)}$ and $e_\lambda = e_{\lambda_1}\cdots e_{\lambda_l}$; the complete homogeneous symmetric functions are $h_n = \sum_{\lvert\mu\rvert = n}m_\mu$ and $h_\lambda = h_{\lambda_1}\cdots h_{\lambda_l}$; and the power sums are $p_n = \sum_i x_i^n$ and $p_\lambda = p_{\lambda_1}\cdots p_{\lambda_l}$.

Definition. For $n\geq l(\lambda)$ and $\lambda$ padded to $n$ parts, the bialternant

$$ s_\lambda(x_1,\dots,x_n) \;=\; \frac{\det\bigl(x_i^{\lambda_j+n-j}\bigr)_{i,j=1}^{n}}{\det\bigl(x_i^{n-j}\bigr)_{i,j=1}^{n}} $$

is a symmetric polynomial in the $n$ variables, and it is stable under the inclusion of one further variable set to zero; the resulting element $s_\lambda$ of $\Lambda$ is the Schur function of $\lambda$, the quotient being the Weyl character formula for the irreducible representation of highest weight $\lambda$ in the type $A$ root system, read as a symmetric polynomial. The Kostka number $K_{\lambda\mu}$ is defined by $s_\lambda = \sum_\mu K_{\lambda\mu}m_\mu$.

Theorem (the fundamental theorem). The ring $\Lambda$ is a polynomial ring over $\mathbb{Z}$: $\Lambda = \mathbb{Z}[e_1,e_2,e_3,\dots] = \mathbb{Z}[h_1,h_2,h_3,\dots]$. Each of the families $\{m_\lambda\}$, $\{e_\lambda\}$, $\{h_\lambda\}$ and $\{s_\lambda\}$, indexed by the partitions, is a $\mathbb{Z}$-basis of $\Lambda$; the family $\{p_\lambda\}$ is a basis of $\Lambda_{\mathbb{Q}}$ but not of $\Lambda$; and the Kostka matrix $(K_{\lambda\mu})$ is unitriangular with respect to the dominance order, so that $s_\lambda = m_\lambda + \sum_{\mu<\lambda}K_{\lambda\mu}m_\mu$ and $K_{\lambda\lambda} = 1$, and the $K_{\lambda\mu}$ are non-negative integers counting the semistandard Young tableaux of shape $\lambda$ and content $\mu$.

Proof. The polynomial ring statement is the standard fundamental theorem of symmetric polynomials: every symmetric function is a polynomial in the elementary ones, and there are no algebraic relations among $e_1,e_2,\dots$, since the $e_i$ are algebraically independent in each finite number of variables and the independence is stable under the inverse limit. The triangularity of the Kostka matrix follows from the bialternant: expanding the two determinants, the highest monomial appearing in $s_\lambda$ is the orbit of $\lambda$ itself with coefficient one, and the remaining monomials have strictly lower dominant weight. The combinatorial interpretation of the $K_{\lambda\mu}$ is the standard tableau description of the Schur functions.

Example. In low degrees, $e_1 = h_1 = p_1 = m_{(1)} = s_{(1)}$; $e_2 = m_{(1,1)}$, $h_2 = m_{(2)}+m_{(1,1)}$, $s_{(2)} = h_2$; and $s_{(2,1)} = m_{(2,1)}+2m_{(1,1,1)}$, the coefficient $2$ being the smallest Kostka number exceeding one and the first sign that the change of basis from the monomial to the Schur basis is not unitriangular with all off-diagonal entries zero.

The Hall Inner Product and the Schur Basis

Definition. The Hall inner product on $\Lambda_{\mathbb{Q}}$ is the symmetric bilinear form $\langle-,-\rangle$ for which the power sums are orthogonal,

$$ \langle p_\lambda,p_\mu\rangle = \delta_{\lambda\mu}\,z_\lambda, \qquad z_\lambda = \prod_{i\geq1}i^{m_i(\lambda)}\,m_i(\lambda)! . $$

Theorem. With respect to the Hall inner product, the Schur functions are an orthonormal $\mathbb{Z}$-basis, $\langle s_\lambda,s_\mu\rangle = \delta_{\lambda\mu}$; the monomial and the complete homogeneous symmetric functions are dual bases, $\langle m_\lambda,h_\mu\rangle = \delta_{\lambda\mu}$; and the Cauchy identity

$$ \prod_{i,j}\frac{1}{1-x_iy_j} \;=\; \sum_\lambda h_\lambda(x)\,m_\lambda(y) \;=\; \sum_\lambda s_\lambda(x)\,s_\lambda(y) $$

holds, the two forms of the identity being exchanged by the duality of the two bases. The Kostka matrix is the transition matrix between the Schur and the monomial bases, and it is the matrix of the inner products $\langle h_\mu,s_\lambda\rangle$.

Proof (outline). The orthonormality of the Schur basis is the classical statement that the Schur functions are the characters of the irreducible polynomial representations of the general linear group with the normalisation of the Haar integral, which over $\mathbb{Z}$ becomes the statement that the scalar product of two Schur functions is the number of tableaux condition, computed by the determinant formula $\langle s_\lambda,s_\mu\rangle = \delta_{\lambda\mu}$; the dual-basis statement follows from the triangularity of the Kostka matrix and the identity $\langle m_\lambda,h_\mu\rangle = \delta_{\lambda\mu}$, and the Cauchy identity is the formal expansion of the product $\prod_{i,j}(1-x_iy_j)^{-1}$ in symmetric functions in either of the two ways. The analytic normalisation of the integral is the measure-theoretic statement that belongs to Part III and is not used; the identity above is proved combinatorially from the expansion of the geometric series.

Proposition. The Hall inner product is the $t = 0$ case of a one-parameter family of inner products, and it is the form with respect to which the Schur basis is orthonormal; it is positive definite on $\Lambda_{\mathbb{R}}$, a statement that itself requires the real ordering and belongs to the theory of the unitary group in Part III.

The Hall–Littlewood Polynomials

Definition. Let $n\geq l(\lambda)$ and let $\lambda$ be padded to $n$ parts by zeros. Set

$$ v_\lambda(t) \;=\; \prod_{i\geq0}\prod_{j=1}^{m_i(\lambda)}\frac{1-t^j}{1-t}, $$

the product over all values $i$ occurring among $\lambda_1,\dots,\lambda_n$, the value $0$ included, with $m_i$ the multiplicity of $i$. The Hall–Littlewood polynomial is

$$ P_\lambda(x_1,\dots,x_n;t) \;=\; \frac{1}{v_\lambda(t)}\sum_{w\in S_n}w\left(x_1^{\lambda_1}\cdots x_n^{\lambda_n}\prod_{i

the sum over the symmetric group acting by permutation of the variables; the expression is independent of $n$ for $n\geq l(\lambda)$ once the parts are padded by zeros, and it defines an element $P_\lambda(x;t)\in\Lambda_{\mathbb{Z}[t]}$.

Remark (the normalisation). The factor $v_\lambda(t)$ is what makes the definition independent of the number $n$ of variables used to present it: the sum over $S_n$ counts the orbit of the exponent vector $\lambda$ with multiplicity the order of the stabiliser, and the multiplicity of the part $0$, that is the number of padded zeros, enters the normalisation. Omitting the zeros from $v_\lambda(t)$ would multiply the polynomial by the factor coming from the stabiliser of the zero part; the two cases $n = l(\lambda)$ and $n>l(\lambda)$ thus differ by exactly that factor, which the definition above cancels.

Theorem (the two limiting cases). The Hall–Littlewood polynomials satisfy

$$ P_\lambda(x;0) = s_\lambda(x), \qquad P_\lambda(x;1) = m_\lambda(x) . $$

Proof. At $t = 0$ the factor $\prod_{i

$$ \sum_{w\in S_n}w\left(x^{\lambda}\prod_{i

with $\delta = (n-1,\dots,1,0)$, because the numerator $\prod_{i

Corollary. The Hall–Littlewood polynomial of the partition $(n)$ of one part is the interpolation between $h_n$ and $m_{(n)}$, and the polynomial of $(1^n)$ is the elementary symmetric function $e_n$ for every $t$; more generally $P_\lambda(x;t)$ has coefficients in $\mathbb{Z}[t]$ and its constant term is $s_\lambda$ and its value at $t = 1$ is $m_\lambda$.

Proposition (the elementary generating function). The elementary symmetric functions are generated by $\sum_{n\geq0}e_n(x)u^n = \prod_i(1+x_iu)$; since $P_{(1^n)}(x;t) = e_n$, the one-column Hall–Littlewood polynomials have the generating function $\sum_{n\geq0}P_{(1^n)}(x;t)u^n = \prod_i(1+x_iu)$ for every $t$. In a single variable every basis of $\Lambda$ agrees, so that $\sum_{n\geq0}P_{(n)}(x;t)u^n = (1-xu)^{-1}$ and $P_{(n)}(x;t) = x^n$ there.

Proof. The identity for $e_n$ is the expansion of the finite product, in which the coefficient of $u^n$, symmetrised, is the sum of the squarefree monomials of degree $n$, that is $e_n$; the one-column case follows from $P_{(1^n)} = e_n$, and the one-variable statement is the observation that the symmetric group is trivial there and that $v_{(n)}(t) = 1$.

The $t$-Deformed Inner Product and Orthogonality

Definition. The $t$-deformed Hall inner product on $\Lambda_{\mathbb{Q}(t)}$ is the bilinear form $\langle-,-\rangle_t$ with the power sums orthogonal and

$$ \langle p_\lambda,p_\mu\rangle_t \;=\; \delta_{\lambda\mu}\,z_\lambda\prod_{i=1}^{l(\lambda)}\frac{1-t^{\lambda_i}}{1-t}, $$

so that at $t = 0$ it is the Hall inner product and at $t = 1$ the factor becomes $\lambda_i$, the product being the function $z_\lambda\prod_i\lambda_i$.

Theorem (orthogonality, standard). The Hall–Littlewood polynomials are pairwise orthogonal for the $t$-deformed form,

$$ \langle P_\lambda(\,\cdot\,;t),P_\mu(\,\cdot\,;t)\rangle_t = 0 \qquad (\lambda\neq\mu), $$

the form is non-degenerate on $\Lambda_{\mathbb{Q}(t)}$ and the $P_\lambda$ therefore form an orthogonal basis. The values $\langle P_\lambda,P_\lambda\rangle_t$ are explicit rational functions of $t$, products over the multiplicities of the parts of $\lambda$, computed in the standard reference; only the vanishing of the off-diagonal inner products is used below, and the normalisation of the form is the one for which the power sums are orthogonal with $\langle p_r,p_r\rangle_t = r\frac{1-t^r}{1-t}$.

Proof (outline). The standard proof computes the inner products of the symmetrised expressions by transporting the sum over $S_n$ to a sum over the symmetric group of inner products of monomials, where the $t$-deformed form has the triangular matrix in the dominance order; the off-diagonal entries vanish by the orthogonality of distinct orbits, and the diagonal entries are the products displayed.

Theorem (Kostka–Foulkes, standard). The transition between the Schur and the Hall–Littlewood bases has coefficients

$$ s_\lambda(x) = \sum_{\mu}K_{\lambda\mu}(t)\,P_\mu(x;t), $$

the Kostka–Foulkes polynomials, with $K_{\lambda\mu}(t)\in\mathbb{N}[t]$, with $K_{\lambda\mu}(t) = 0$ unless $\mu\leq\lambda$ in the dominance order, with $K_{\lambda\lambda}(t) = 1$, and with the two limiting values

$$ K_{\lambda\mu}(0) = \delta_{\lambda\mu}, \qquad K_{\lambda\mu}(1) = K_{\lambda\mu}, $$

the first because $P_\mu(x;0) = s_\mu$ and the second because $P_\mu(x;1) = m_\mu$ and $s_\lambda = \sum_\mu K_{\lambda\mu}m_\mu$. Equivalently the coefficients are the generating functions of the charge statistic of Lascoux–Schützenberger on the semistandard tableaux, the normalisation of the statistic being fixed by the two limiting values above.

Proof (outline). The existence and the integrality of the transition coefficients follow from the triangularity of the two bases with respect to the dominance order and from the integrality of the change of basis between any two of the standard bases with the same triangularity; the non-negativity is the theorem of Lascoux–Schützenberger, whose proof exhibits the coefficients as the generating polynomials of the charge statistic over the semistandard tableaux. The geometric interpretation of the polynomials as the Poincaré polynomials of a graded piece of the cohomology of a Springer fibre belongs to Part II, where the algebraic variety is available.

Example. For $\lambda = (2,1)$ in three variables the monomial expansion is $P_{(2,1)}(x;t) = m_{(2,1)}+u(t)m_{(1,1,1)}$ with the polynomial $u(t) = (1-t)(2+t) = 2-t-t^2$, whose values at the two ends are $u(0) = 2$ and $u(1) = 0$, since $P_{(2,1)}(x;0) = s_{(2,1)} = m_{(2,1)}+2m_{(1,1,1)}$ and $P_{(2,1)}(x;1) = m_{(2,1)}$. Both were checked by explicit computation at $(x_1,x_2,x_3) = (2,3,5)$: the value at $t = 0$ is $280$, which equals $s_{(2,1)} = h_1h_2-h_3$ at that point, the value at $t = 1$ is $220 = m_{(2,1)}$, and the value at $t = 2$ is $100$ with $u(2) = -4$, so that $100+6\cdot30 = 280$ recovers $s_{(2,1)}$ there. The transition coefficients of the theorem in this case are $K_{(2,1),(2,1)}(t) = 1$ and $K_{(2,1),(1,1,1)}(t) = t+t^2$, the second determined by $K_{(2,1),(1,1,1)}(t) = 2-u(t)$ and evaluated as $0$ at $t = 0$ and as $2 = K_{(2,1),(1,1,1)}$ at $t = 1$, so both limiting values of the theorem are visible in the smallest case with a non-trivial Kostka number.

The Macdonald Polynomials

Definition. The $(q,t)$-deformed Hall inner product on $\Lambda_{\mathbb{Q}(q,t)}$ is the bilinear form with

$$ \langle p_r,p_s\rangle_{q,t} = \delta_{rs}\,r\,\frac{1-q^r}{1-t^r}, $$

so that at $q = 0$ it is a renormalisation of the $t$-deformed form of the previous section, and at $q = t$ the factor $\frac{1-q^r}{1-t^r}$ is identically one and the form is the classical Hall form.

Definition. The Macdonald polynomials $P_\lambda(x;q,t)$ are the symmetric functions characterised by the two conditions:

(triangularity) $P_\lambda = m_\lambda + \sum_{\mu<\lambda}u_{\lambda\mu}(q,t)\,m_\mu$ with $u_{\lambda\mu}\in\mathbb{Q}(q,t)$;
(orthogonality) $\langle P_\lambda,P_\mu\rangle_{q,t} = 0$ whenever $\lambda\neq\mu$.

Theorem (Macdonald, standard). For every partition $\lambda$ there is a unique symmetric function $P_\lambda(x;q,t)$ satisfying the two conditions, and the assignment $\lambda\mapsto P_\lambda$ is a basis of $\Lambda_{\mathbb{Q}(q,t)}$ indexed by the partitions, triangular with respect to the monomial basis.

Proof (outline). The bilinear form $\langle-,-\rangle_{q,t}$ is non-degenerate, and the conditions amount to the statement that $P_\lambda$ is the unique element of the form $m_\lambda+\sum_{\mu<\lambda}c_\mu m_\mu$ that is orthogonal to the span of the $m_\mu$ with $\mu<\lambda$; the existence and the uniqueness are the standard orthonormalisation argument with respect to a non-degenerate form, applied in the order of the dominance order, and the coefficients are computed by the inversion of the (triangular) matrix of inner products.

Theorem (specialisations, standard). The Macdonald polynomials specialise as follows:

$$ t = 0:\ P_\lambda(x;q,0)\ \text{ is a scalar multiple of the Hall–Littlewood polynomial } P_\lambda(x;q), $$

$$ q = t:\ P_\lambda(x;t,t)\ \text{ is a scalar multiple of the Schur function } s_\lambda, $$

$$ q = t^{\alpha},\ t\to1:\ P_\lambda\ \text{tends to the Jack polynomial } P^{(\alpha)}_\lambda . $$

The $q = t$ specialisation reduces the deformed form to the Hall form, since the factor $\frac{1-q^r}{1-t^r}$ becomes $1$; the $t = 0$ specialisation makes the $(q,t)$-form a renormalisation of the $q$-deformed form of the previous section, so that only the normalisation, and not the orthogonality, changes; and the Jack limit is the one-parameter family obtained by the substitution $q = t^{\alpha}$ and the limit $t\to1$ with $\alpha$ fixed. The degenerate cases $q = 0$ and $t = 1$ give the $q$-Whittaker functions and the monomial basis respectively, again up to normalisation.

Theorem (Macdonald positivity, Haiman). The change of basis

$$ P_\lambda(x;q,t) = \sum_\mu K_{\lambda\mu}(q,t)\,s_\mu(x) $$

has coefficients $K_{\lambda\mu}(q,t)\in\mathbb{N}[q,t]$, the $(q,t)$-Kostka polynomials; at $q = t$ the form degenerates to the Hall form, the polynomials become the Schur functions, and therefore $K_{\lambda\mu}(t,t) = 0$ for $\lambda\neq\mu$; at $t = 0$ the coefficients are, up to the normalisation of the polynomials, the entries inverse to the Kostka–Foulkes matrix of the previous section, and at $q = 0$ they are the $q$-Kostka polynomials of the Hall–Littlewood theory.

The positivity was conjectured by Macdonald and proved by Haiman from the geometry of the Hilbert scheme of points of the plane: the polynomials are the Poincaré polynomials of the fibres of a certain graded character, and the geometric proof uses the cohomology of a variety, hence belongs to Part II. The algebraic content, the non-negativity of the coefficients, is stated here and the geometric proof is deferred.

Example. For $\lambda = (1)$ the Macdonald polynomial is $m_{(1)}$, and for $\lambda = (2)$ and $\lambda = (1,1)$ the triangularity together with the orthogonality determines the polynomial from the two inner products $\langle m_{(2)},m_{(2)}\rangle_{q,t}$ and $\langle m_{(2)},m_{(1,1)}\rangle_{q,t}$; the one-row and one-column Macdonald polynomials were computed by Macdonald in closed form. The first case in which the two parameters enter a genuinely two-dimensional coefficient is $\lambda = (2,1)$, where the coefficient of $m_{(1,1,1)}$ is a rational function of $q$ and $t$ that reduces to the constant $2$ at $q = 0$.

The Hecke-Theoretic Interpretation

The Hall–Littlewood and Macdonald polynomials are not merely the solutions of an orthogonality problem: they are the eigenfunctions of an algebra of operators coming from the Hecke algebra of the previous article, and this is the reason they belong to the present category.

Definition. Let $n\geq1$ and let $\Lambda^{(n)}$ be the span of the elements of $\Lambda$ of degree $n$. The Hecke operators $U_r$, $r\geq1$, are the linear endomorphisms of $\Lambda$ determined by

$$ U_r(p_\mu) = \sum_{\nu} c^{\mu}_{r,\nu}(t)\,p_\nu , $$

with structure constants read off from the action of the Hecke algebra $H_t(S_n)$ of Hecke Algebras on the polynomial ring $k[x_1,\dots,x_n]$ and its coinvariant quotient, the symmetric group being the type $A$ Weyl group; the operators commute and generate a commutative subalgebra of the endomorphisms of $\Lambda^{(n)}$.

Theorem (standard). The Hall–Littlewood polynomials $P_\lambda(x;t)$ with $\lvert\lambda\rvert = n$ are the simultaneous eigenfunctions of the Hecke operators $U_r$ on $\Lambda^{(n)}$, and the eigenvalues are obtained by specialising the appropriate symmetric polynomials to the multiset $\{t^{\lambda_i+n-i}\}_{i=1}^{n}$ of $t$-powers attached to $\lambda$; in the limit $t = 0$ the operators degenerate to the multiplication operators of the classical theory, whose eigenvectors are the Schur functions. Likewise the Macdonald polynomials $P_\lambda(x;q,t)$ are simultaneous eigenfunctions of the Cherednik operators, the operator family of the double affine Hecke algebra, and their eigenvalues are obtained by specialising the corresponding symmetric polynomials to the multiset $\{q^{\lambda_i}t^{n-i}\}_{i=1}^{n}$, up to the standard normalisation.

Proof (outline). The Hecke algebra $H_t(S_n)$ of Hecke Algebras acts on the polynomial ring $k[x_1,\dots,x_n]$ through the operators $\pi_i = T_i$ in the standard "Macdonald representation", and the action preserves the symmetric polynomials after the appropriate symmetrisation; the simultaneous diagonalisation of the commuting operators is the algebraic form of the orthogonality of the Hall–Littlewood polynomials with respect to the $t$-deformed form, and the eigenvalues are computed on the monomial basis by the triangularity. The Cherednik operator statement is the affine analogue; the double affine Hecke algebra and its operator calculus are the objects whose analytic and geometric development belongs to Part III, and the algebraic eigenfunction statement is recorded here with that boundary.

Corollary. The $t$-deformed inner product of the previous section is the form with respect to which the Hecke algebra action of type $A$ is self-adjoint, and the property of being an orthogonal basis for that form is equivalent to the eigenfunction property for the Hecke operators. The two constructions of the Hall–Littlewood polynomials — the symmetrisation formula and the eigenfunction property — therefore give the same family, and the identification is the algebraic reason for the interpolation between the Schur and the monomial bases: at $t = 0$ the Hecke algebra degenerates to the group algebra of the symmetric group, whose characters are the Schur functions, and at $t = 1$ the Hecke algebra degenerates to the monoid algebra of the $0$-Hecke monoid, whose action on the monomial basis is the trivial one.

Summary

Let $\Lambda$ be the ring of symmetric functions, the inverse limit of the rings of symmetric polynomials in $n$ variables, with the bases given by the monomial symmetric functions $m_\lambda$, the elementary functions $e_\lambda$, the complete homogeneous functions $h_\lambda$, the power sums $p_\lambda$ and the Schur functions $s_\lambda$; the Kostka numbers $K_{\lambda\mu}$ are the transition coefficients $s_\lambda = \sum_\mu K_{\lambda\mu}m_\mu$, with $K_{\lambda\lambda} = 1$ and $K_{\lambda\mu}\in\mathbb{N}$. The Hall inner product is the form with the power sums orthogonal and $\langle p_\lambda,p_\mu\rangle = \delta_{\lambda\mu}z_\lambda$, with respect to which the Schur functions are orthonormal and the monomial and complete homogeneous functions are dual, and the Cauchy identity holds. The Hall–Littlewood polynomials are defined by the symmetrisation

$$ P_\lambda(x;t) = \frac{1}{v_\lambda(t)}\sum_{w\in S_n}w\left(x^\lambda\prod_{i

they satisfy $P_\lambda(x;0) = s_\lambda$ and $P_\lambda(x;1) = m_\lambda$, both specialisations verified by explicit computation in small cases, the transition to the Schur basis is given by the Kostka–Foulkes polynomials, $s_\lambda = \sum_\mu K_{\lambda\mu}(t)P_\mu(x;t)$ with $K_{\lambda\mu}(t)\in\mathbb{N}[t]$, $K_{\lambda\mu}(0) = \delta_{\lambda\mu}$ and $K_{\lambda\mu}(1) = K_{\lambda\mu}$, and they are pairwise orthogonal for the $t$-deformed form with $\langle p_\lambda,p_\mu\rangle_t = \delta_{\lambda\mu}z_\lambda\prod_i\frac{1-t^{\lambda_i}}{1-t}$. The Macdonald polynomials $P_\lambda(x;q,t)$ are characterised by the triangularity $P_\lambda = m_\lambda+\sum_{\mu<\lambda}u_{\lambda\mu}m_\mu$ and the orthogonality for the $(q,t)$-form $\langle p_r,p_s\rangle_{q,t} = \delta_{rs}r\frac{1-q^r}{1-t^r}$; they specialise to the Hall–Littlewood polynomials at $q = 0$, to the Schur functions at $q = t$, to the Jack polynomials $P^{(\alpha)}_\lambda$ in the limit $q = t^{\alpha}$, $t\to1$, and their expansion in the Schur basis has non-negative integral polynomial coefficients $K_{\lambda\mu}(q,t)$, the $(q,t)$-Kostka polynomials of the Macdonald positivity theorem, proved by Haiman from the geometry of the Hilbert scheme, a Part II argument. Finally the Hall–Littlewood and Macdonald polynomials are the simultaneous eigenfunctions of the Hecke operators of type $A$ and of the Cherednik operators respectively, the latter belonging to the double affine Hecke algebra whose analytic theory is Part III; this is the sense in which the two families belong to the theory of the Hecke algebra of Hecke Algebras and to the present category.

Summary of Notation

Symbol	Meaning
$\Lambda$, $\Lambda_R$	ring of symmetric functions; inverse limit of the symmetric polynomials
$\lambda$, $\lvert\lambda\rvert$, $l(\lambda)$	partition, its size, its number of parts
$m_i(\lambda)$	multiplicity of the part $i$
$m_\lambda,e_\lambda,h_\lambda,p_\lambda,s_\lambda$	monomial, elementary, complete, power-sum, Schur bases
$K_{\lambda\mu}$	Kostka number, $s_\lambda = \sum_\mu K_{\lambda\mu}m_\mu$
$z_\lambda = \prod_i i^{m_i}m_i!$	order of the centraliser, normalising the power sums
$\langle-,-\rangle$	Hall inner product, $\langle p_\lambda,p_\mu\rangle = \delta_{\lambda\mu}z_\lambda$
$P_\lambda(x;t)$, $v_\lambda(t)$	Hall–Littlewood polynomial and its normalising factor
$K_{\lambda\mu}(t)$	Kostka–Foulkes polynomial, charge statistic
$\langle-,-\rangle_t$	$t$-deformed form, $\langle p_\lambda,p_\mu\rangle_t = \delta_{\lambda\mu}z_\lambda\prod_i\frac{1-t^{\lambda_i}}{1-t}$
$P_\lambda(x;q,t)$, $K_{\lambda\mu}(q,t)$	Macdonald polynomial and $(q,t)$-Kostka polynomial
$\langle-,-\rangle_{q,t}$	$(q,t)$-form, $\langle p_r,p_s\rangle_{q,t} = \delta_{rs}r\frac{1-q^r}{1-t^r}$
$P^{(\alpha)}_\lambda$, $W_\lambda$	Jack polynomial, $q$-Whittaker function
$U_r$	Hecke operators on $\Lambda$