Schur polynomials

schur_polynomial(lambda::Partition{T}, n::Int=length(lambda)) where T<:Integer
schur_polynomial(R::ZZMPolyRing, lambda::Partition{T}, n::Int=length(lambda)) where T<:Integer

Return the Schur polynomial $s_λ(x_1,x_2,...,x_n)$ in n variables.

If R is not given, the Schur polynomial will be over polynomial_ring(ZZ,n).

Examples

julia> R,x = polynomial_ring(ZZ, ["a","b","c"]);

julia> schur_polynomial(R, Partition([2,1]))
a^2*b + a*b^2

julia> schur_polynomial(R, Partition([2,1]), 3)
a^2*b + a^2*c + a*b^2 + 2*a*b*c + a*c^2 + b^2*c + b*c^2

julia> schur_polynomial(Partition([2]))
x1^2

Algorithm

We use two different Algorithms, depending on the size of the input. The Combinatorial Algorithm is used for Partitions of small integers, or if $n ≥ 10$. In the other cases we use Cauchy's bialternant formula.

Combinatorial Algorithm

\[s_λ:=∑_T x_1^{m_1}…x_n^{m_n}\]

where the sum is taken over all semistandard tableaux $T$ of shape $λ$, and $m_i$ gives the weight of $i$ in $T$.

Cauchy's bialternant formula

\[s_\lambda(x_1,\dots,x_n) = \prod_{1\leq i < j \leq n} (x_i-x_j)^{-1} \begin{vmatrix} x_1^{λ_1+n-1} & x_2^{λ_1+n-1} & … & x_n^{λ_1+n-1} \\ x_1^{λ_2+n-2} & x_2^{λ_2+n-2} & … & x_n^{λ_2+n-2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ x_1^{λ_n} & x_2^{λ_n} & … & x_n^{λ_n} \end{vmatrix}\]