Wreath Macdonald polynomials

The existence, integrality and positivity of wreath Macdonald polynomials has been conjectured by Haiman [Hai02] and proved by Bezrukavnikov and Finkelberg [BF14]. Here we have implemented an algorithm computing the wreath Macdonald polynomials as defined in the survey by Orr and Shimozono on this topic [OS23].

Take $r$ and $n$ two integers and consider the complex reflection group $G(r,1,n):= (\mathbb{Z}/r\mathbb{Z})^n \rtimes \mathfrak{S}_n$ with its natural $n$-dimensional reflection representation $\mathfrak{h}_n$. Let $\Lambda^i\mathfrak{h}_n^*$ denote the exterior power of $\mathfrak{h}_n^*$. The irreducible representations over $\mathbb{C}$ of $G(r,1,n)$ are indexed by the $r$-multipartitions of size $n$. For $\lambda^{\bullet}$ such a multipartition, let us denote by $V_{\lambda^{\bullet}}$ the associated irreducible representation. If $V$ is a representation of $G(r,1,n)$, let $\left[V\right]$ denote the class of $V$ in $K_{G(r,1,n)}$, the Grothendieck ring of $G(r,1,n)$.
We denote by $\mathbb{K}$ the field $\mathbb{Q}(q,t)$, where $q$ and $t$ are indeterminates over $\mathbb{Q}$. Let $R$ denote the ring of symmetric functions over $\mathbb{K}$, see $\S I.2$ in [Mac15]. The ring of $r$-multisymmetric functions is the $r$-fold tensor product of $R$ over $\mathbb{K}$. If one equips $\bigoplus_{n \in \mathbb{Z}_{\geq 0}}{K_{G(r,1,n)}}$ with the induction product, then in light of Theorem $2.3$ in [Wen19], we identify the ring of $r$-multisymmetric functions and $\bigoplus_{n \in \mathbb{Z}_{\geq 0}}{K_{G(r,1,n)}}$.
For $\lambda$ a partition, denote respectively by $\mathrm{core}_r(\lambda)$ and $\mathrm{quot}_r(\lambda)$ the $r$-core and $r$-quotient of $\lambda$, see $I.3$ in [Ols93]. Moreover, let us equip the set of partitions with $\leq$, the dominance order. For an element $\omega$ of $\mathfrak{S}_r$ and a partition $\lambda$, define $\omega.\lambda$ to be the partition with the same $r$-core as $\lambda$ and $r$-quotient equal to $\omega.\mathrm{quot}_r(\lambda)$ where $\omega$ acts by permuting the $r$ partitions. We define the order $\leq_{\omega}$ as follows. If $\lambda$ and $\mu$ are two partitions, then $\lambda \leq_{\omega} \mu$ if $\omega.\lambda \leq \omega.\mu$. We now give a simple characterization of the wreath Macdonald polynomials.

For each partition $\lambda$ such that $|\mathrm{quot}_r(\lambda)|=n$ and each $\omega \in \mathfrak{S}_r$, the wreath Macdonald polynomial $H^{\omega}_{\lambda}$ is the $r$-multisymmetric function uniquely characterized by

• $H^{\omega}_{\lambda} \otimes \sum_{i=0}^n(-q)^i\left[\Lambda^i\mathfrak{h}_n^*\right] \in \bigoplus_{\mu \geq_{\omega} \lambda, \mathrm{core}_r(\mu)=\mathrm{core}_r(\lambda)}{\mathbb{K}\left[V_{\mathrm{quot}(\mu)}\right]}$,
• $H^{\omega}_{\lambda} \otimes \sum_{i=0}^n(-t)^{-i}\left[\Lambda^i\mathfrak{h}_n^*\right] \in \bigoplus_{\mu \leq_{\omega} \lambda, \mathrm{core}_r(\mu)=\mathrm{core}_r(\lambda)}{\mathbb{K}\left[V_{\mathrm{quot}(\mu)}\right]}$,
• $\langle H^{\omega}_{\lambda},[\mathrm{triv}]\rangle = 1$.

Remark that when $r=1$, wreath Macdonald polynomials are equal to the Haiman-Macdonald polynomials, used to prove the Macdonald positivity conjecture.

These polynomials, apart from generalizing the Haiman-Macdonald polynomials and giving access to new combinatorics, have a geometric counterpart. Denote by $\mathbb{T}$ the maximal diagonal torus of $\mathrm{GL}_2(\mathbb{C})$. To be more precise, Bezrukavnikov and Finkelberg prove that $H^{\omega}_{\lambda}$ can be realized as the bigraded $G(r,1,n)$ Frobenius character of the fiber of a wreath Procesi bundle (see [Los18]) at the $\mathbb{T}$-fixed point associated with $\mathrm{quot}_r(\lambda)$.

Finally, the wreath Macdonald polynomials can be interpreted as the eigenbasis of explicit vertex operators, see [Wen19].

In our implementation, wreath Macdonald polynomials depend on two parameters. The first parameter is an $r$-multipartition of $n$ . The second parameter is an element of the affine Weyl group of type $A^{(1)}_{r-1}$ which is isomorphic to the semi-direct product of the finite Weyl group of type $A_{r-1}$ (the symmetric group on $r$ letters) and of the coroot lattice of type $A_{r-1}$, where the finite Weyl group acts by permutation. The element of the coroot lattice is given in the canonical basis. It is then the sublattice of $\mathbb{Z}^r$ of elements summing up to zero.
It is equivalent to the data of a partition $\lambda$ such that $|\mathrm{quot}_r(\lambda)|=n$ and an element $\omega \in \mathfrak{S}_r$. Indeed the $\mathrm{quot}_r(\lambda)$ is an $r$-multipartition of $n$. Moreover, the set of $r$-cores is in bijection with the coroot lattice of type $A_{r-1}$ (note that this bijection is explicit using abaci). Finally, a partition is entirely determined by its $r$-core and $r$-quotient, see Proposition 3.7 in [Ols93].

wreath_macdonald_polynomial(lbb::Multipartition,
                            wperm::PermGroupElem,
                            coroot::Vector{Int};
                            parent::MPolyRing{<:QQAbFieldElem}=
                            polynomial_ring(abelian_closure(QQ)[1], [:q,:t];cached=true)[1])

julia> wreath_macdonald_polynomial(multipartition([[1],[],[]]),cperm(1:3),[0,1,-1])
[q   q^2   1]

wreath_macdonald_polynomials(n::Int,
                             r::Int,
                             wperm::PermGroupElem,
                             coroot::Vector{Int};
                             parent::MPolyRing{<:QQAbFieldElem}=
                             polynomial_ring(abelian_closure(QQ)[1], [:q,:t];cached=true)[1])

julia> wreath_macdonald_polynomials(1,3,cperm(1:3),[0,1,-1])
[t^2     t   1]
[  q     t   1]
[  q   q^2   1]

Compare the following computation with Example 3.15 in [OS23].

julia> collect(multipartitions(1,3))
3-element Vector{Multipartition{Int64}}:
 Partition{Int64}[[], [], [1]]
 Partition{Int64}[[], [1], []]
 Partition{Int64}[[1], [], []]

julia> wreath_macdonald_polynomials(1,3,cperm(1:3),[0,1,-1])[[3, 2, 1],[3, 2, 1]]
[1   q^2     q]
[1     t     q]
[1     t   t^2]

Contact

Please direct questions about this part of OSCAR to: Raphaël Paegelow.