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]. When $r=1$, wreath Macdonald polynomials are equal to the Haiman-Macdonald polynomials, used to prove the Macdonald positivity conjecture.

Here we have implemented an algorithm computing the wreath Macdonald polynomials as defined in the survey by Orr and Shimozono on this topic [OS23].

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 $A_{r-1}$. 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.

wreath_macdonald_polynomial — Function

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

Given a multipartition lbb of size $n$ and length $r$ and an element of the affine Weyl group of type $A^{(1)}_{r-1}$ (seen as the semi-direct product of the symmetric group on $r$ letters with the coroot lattice of the finite type $A_{r-1}$), this function returns the coefficients of the wreath Macdonald polynomial associated with lbb and the affine Weyl group element in the standard Schur basis of multisymmetric functions. Here is an example of how to use it:

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

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

wreath_macdonald_polynomials — Function

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])

Given two integers n and r and an element of the affine Weyl group of type $A^{(1)}_{r-1}$ (seen as the semi-direct product of the symmetric group on r letters with the coroot lattice of the finite type $A_{r-1}$), this function returns the square matrix of coefficients of the wreath Macdonald polynomials associated with all multipartitions of size n and length r in the standard Schur basis of multisymmetric functions. Each row of this matrix is a wreath Macdonald polynomial. Here is an example of how to use it:

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

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

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]