higherSpechtPolynomial(S,R)
higherSpechtPolynomial(S,standard,R)
higherSpechtPolynomial(p,R)
higherSpechtPolynomial(R)
This methods returns higher Specht polynomials sorted in hash tables depending on the input received.
If the input is just a YoungTableau $S$ of shape $\lambda$ and a PolynomialRing then it calculates the standard tableaux $ST(\lambda)$ and then stores all polynomials $F_T^S$ such that $T \in ST(\lambda)$. The polynomials are stored in a hash table with the filling of $T$ as the key.
The list $ST(\lambda)$ can be provided as an input. This is used to avoid repeating this calculation when this method is called multiple times with the same shape $\lambda$.
This set forms a basis for one of the copies of the Specht module $S^\lambda$.
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If only a partition $\lambda$ and a polynomial ring is given then the method calculates $ST(\lambda)$. Then it calculates all polynomials $F_T^S$ such that $S,T \in ST(\lambda)$.
This is a basis for the isotypical component $X_\lambda$ in the coinvariant algebra of the symmetric group.
The polynomials are stored by calling for each $S \in ST(\lambda) $ the previous method. The output is stored in another hash table with the key being the filling of the tableau $S$.
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Finally if just a polynomial ring $R$ with $n$ elements is provided then the method calculates all higher Specht polynomials for all partitions $\lambda$ of $n$.
The polynomials are calculated by calling the previous method for every partition of $n$ and storing the values in a new hash table with the key being the partition.
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The object higherSpechtPolynomials is a method function with options.