Explicit Pieri formulas for Macdonald spherical functions and polynomials beyond small dominant weights

For an irreducible reduced crystallographic root system $R$ with Weyl group $W$ , the Macdonald spherical functions (generalized Hall–Littlewood polynomials) form a distinguished basis of the $W$ -invariant group algebra $\mathbb{C}[P]^W$ , and their $q$ -deformations are the Macdonald polynomials. Van Diejen and Emsiz (2011) proved explicit Pieri formulas — expansions of the product of a basis element $P_\lambda$ with an explicit generator $E_\omega$ in this basis — but only for small dominant weights $\omega$ satisfying $\langle\omega,\alpha^\vee\rangle\le 2$ for all positive roots $\alpha$ ; the coefficients of terms outside the highest-weight orbit are alternating sums rather than closed factorized expressions, and for the exceptional root systems the small fundamental weights supply only part of a generating system of $\mathbb{C}[P]^W$ . The subsequent literature through 2024 leaves open the extension of the explicit Pieri calculus beyond small dominant weights and, with it, the completion of the exceptional-type Pieri generating systems — equivalently, explicit formulas providing a complete system of generators of the commuting operator algebra, valid uniformly for all root systems.

Background

Let $R$ be an irreducible reduced crystallographic root system in a Euclidean space, with Weyl group $W$ (the finite reflection group generated by the reflections in the roots), root lattice $Q$ , and weight lattice $P$ ; the dominant weights $P^+$ are those on which all positive roots $\alpha\in R^+$ take nonnegative values under the pairing $\langle\lambda,\alpha^\vee\rangle$ , where the coroot is $\alpha^\vee=2\alpha/\langle\alpha,\alpha\rangle$ . The group algebra $\mathbb{C}[P]$ has a natural monomial basis $e_\lambda$ ( $e_\lambda e_\mu=e_{\lambda+\mu}$ ), and $W$ acts on it, so the invariant subalgebra $\mathbb{C}[P]^W$ is a polynomial algebra of rank $n=\mathrm{rank}(R)$ ; it plays the role that the ring of symmetric functions plays in type $A$ . Through the Satake isomorphism — which identifies the spherical Hecke algebra of a $p$ -adic group with the Weyl-invariant algebra of its dual root system — $\mathbb{C}[P]^W$ carries a canonical basis of Macdonald spherical functions $P_\lambda$ ( $\lambda\in P^+$ ), depending on a $W$ -invariant multiplicity parameter $t$ ; in type $A_n$ they are the classical Hall–Littlewood symmetric functions. Their $q$ -deformations are the Macdonald polynomials, the joint eigenbasis of a commuting algebra of $q$ -difference operators (Macdonald operators, constructed uniformly via the double affine Hecke algebra, a deformation of the group algebra of the affine Weyl group that governs the whole theory) that is isomorphic to $\mathbb{C}[P]^W$ .

A Pieri formula expresses multiplication by a ring generator in the distinguished basis — the name comes from the classical Pieri rule for multiplying a Schur function by an elementary symmetric function — and such formulas encode the complete multiplicative structure (structure constants) of the ring. Macdonald's monograph (2003) gives uniform explicit Pieri-type formulas for the minuscule weights ( $\langle\omega,\alpha^\vee\rangle\le 1$ for all $\alpha\in R^+$ ) and the quasi-minuscule weights (the bound $2$ is attained only by the highest root), and it highlights as an open problem the construction of explicit formulas for a complete system of generators of the commuting operator algebra, uniformly for all root systems.

Van Diejen and Emsiz (2011) proved an explicit Pieri formula for $E_{\omega;R}P_\lambda$ (with $E_{\omega;R}$ an explicit triangular generator of leading monomial $m_{\omega;R}$ , the orbit sum of $\omega$ ) valid for all small dominant weights $\omega$ , those with $\langle\omega,\alpha^\vee\rangle\le 2$ for all $\alpha\in R^+$ — a class strictly between minuscule and general. The defining formula for $E_{\omega;R}$ , a signed sum over chains $\mu_1<\cdots<\mu_\ell=\omega$ of dominant weights, is well-formed for every dominant weight $\omega$ ; the smallness hypothesis enters the proofs, not the definition of the generator. The proof degenerates a corresponding formula at the Macdonald-polynomial level, itself obtained from a difference equation and a residue calculus over root-system hyperplanes; the smallness hypothesis enters because for small $\omega$ the set $P^+_\omega=\{\mu\in P^+:\mu\le\omega\}$ (dominance order: $\lambda\ge\mu$ iff $\lambda-\mu\in Q^+$ ) forms a linear chain, which makes the alternating-sum structure of the coefficients tractable. Since all fundamental weights of the classical root systems are small, this yields a complete explicit Pieri generating system for $\mathbb{C}[P]^W$ in types $A$ – $D$ , recovering the classical type- $A$ Pieri formulas as special cases; but for the exceptional types only some fundamental weights are small ( $E_6$ : $\omega_1,\omega_2,\omega_3,\omega_5,\omega_6$ ; $E_7$ : $\omega_1,\omega_2,\omega_6,\omega_7$ ; $E_8$ : $\omega_1,\omega_8$ ; $F_4$ : $\omega_1,\omega_4$ ; $G_2$ : $\omega_1$ , in Bourbaki numbering, i.e. the standard numbering of the Dynkin-diagram tables), so only part of a generating system is obtained — e.g. in type $E_8$ just two explicit generators for a rank- $8$ algebra. Moreover, the structure coefficients attached to weights $\mu<\omega$ outside the highest-weight orbit appear only as alternating sums of products, without a closed factorized or combinatorial form.

Problem Statement

Let $R$ be an irreducible reduced crystallographic root system with Weyl group $W$ , weight lattice $P$ , and dominant cone $P^+$ . The problem is to extend the explicit Pieri calculus of van Diejen–Emsiz (2011) beyond small dominant weights so as to obtain explicit formulas providing a complete system of generators of the commuting operator algebra — equivalently, a complete explicit Pieri generating system of $\mathbb{C}[P]^W$ in the Macdonald spherical function basis, with the corresponding Macdonald-polynomial $q$ -deformation — valid uniformly for all root systems; this is the problem highlighted in Macdonald's monograph (2003) and stated as open by van Diejen–Emsiz (Int. Math. Res. Not. 2011). Concretely, two linked facets: (i) for the dominant weights $\omega\in P^+$ beyond the small class ( $\langle\omega,\alpha^\vee\rangle\le 2$ for all $\alpha\in R^+$ ) that are needed as algebra generators — in particular the fundamental weights that are not small — give an explicit Pieri formula for the expansion of $E_\omega P_\lambda$ in the spherical-function basis, together with the corresponding $q$ -deformed formula for the Macdonald polynomials and the associated commuting difference-operator construction; (ii) for the exceptional root systems $E_6,E_7,E_8,F_4,G_2$ , thereby complete the explicit Pieri generating systems of $\mathbb{C}[P]^W$ (in type $E_8$ , for instance, only $\omega_1$ and $\omega_8$ are currently explicit, for a rank- $8$ algebra). Facets (i)–(ii) are linked: (ii) is the type-by-type completion of (i), and both concern the same multiplicative structure (structure constants) of $\mathbb{C}[P]^W$ .

References

J. F. van Diejen and E. Emsiz, "Pieri formulas for Macdonald's spherical functions and polynomials", Mathematische Zeitschrift 269 (2011) 281–292, DOI 10.1007/s00209-010-0727-0, https://doi.org/10.1007/s00209-010-0727-0 (preprint: arXiv:1009.4482, titled "A Pieri formula for Macdonald's spherical functions and polynomials", https://arxiv.org/abs/1009.4482)
J. F. van Diejen and E. Emsiz, "A generalized Macdonald operator", International Mathematics Research Notices 2011 (2011) 3560–3574, DOI 10.1093/imrn/rnq233, https://doi.org/10.1093/imrn/rnq233 (preprint: arXiv:1009.4486, https://arxiv.org/abs/1009.4486)
J. F. van Diejen and E. Emsiz, "Unitary representations of affine Hecke algebras related to Macdonald spherical functions", Journal of Algebra 354 (2012) 180–210, DOI 10.1016/j.jalgebra.2012.01.005, https://doi.org/10.1016/j.jalgebra.2012.01.005 (preprint: arXiv:1209.3291, https://arxiv.org/abs/1209.3291)
J. F. van Diejen, "A Pieri formula for the characters of complex simple Lie algebras", Transformation Groups 29 (2024) 47–75, DOI 10.1007/s00031-022-09735-7, https://doi.org/10.1007/s00031-022-09735-7
J. F. van Diejen, E. Emsiz, and I. N. Zurrián, "Affine Pieri rule for periodic Macdonald spherical functions and fusion rings", Advances in Mathematics 392 (2021) 108027, DOI 10.1016/j.aim.2021.108027, https://doi.org/10.1016/j.aim.2021.108027 (preprint: arXiv:2305.01931, https://arxiv.org/abs/2305.01931)
M. Yip, "A Littlewood–Richardson rule for Macdonald polynomials", Mathematische Zeitschrift 272 (2012) 1259–1290, DOI 10.1007/s00209-012-0986-z, https://doi.org/10.1007/s00209-012-0986-z
I. G. Macdonald, "Affine Hecke Algebras and Orthogonal Polynomials", Cambridge University Press, Cambridge, 2003, DOI 10.1017/cbo9780511542824, https://doi.org/10.1017/cbo9780511542824
W. Baratta, "Pieri-type formulas for nonsymmetric Macdonald polynomials", International Mathematics Research Notices 2009 (2009) No. 15, 2829–2854, DOI 10.1093/imrn/rnp034, https://doi.org/10.1093/imrn/rnp034 (preprint: arXiv:0806.2695, https://arxiv.org/abs/0806.2695)