Math.Algebra.Jack.SymmetricPolynomials

A Jack polynomial can have a very long expression which can be considerably reduced if the polynomial is written in the basis formed by the monomial symmetric polynomials instead. This is the motivation of this module.

Documentation

isSymmetricSpray :: (C a, Eq a) => Spray a -> Bool

Checks whether a spray defines a symmetric polynomial; this is useless for Jack polynomials because they always are symmetric, but this module contains everything needed to build this function and it can be useful in another context

msPolynomial

Arguments

:: (C a, Eq a)

=> Int

number of variables

-> Partition

integer partition

-> Spray a

Monomial symmetric polynomials

>>> putStrLn $ prettySpray' (msPolynomial 3 [2, 1])
(1) x1^2.x2 + (1) x1^2.x3 + (1) x1.x2^2 + (1) x1.x3^2 + (1) x2^2.x3 + (1) x2.x3^2

msCombination :: C a => Spray a -> Map Partition a

Symmetric polynomial as a linear combination of monomial symmetric polynomials

prettySymmetricNumSpray :: (Num a, Ord a, Show a, C a) => Spray a -> String

Prints a symmetric spray as a linear combination of monomial symmetric polynomials

>>> putStrLn $ prettySymmetricNumSpray $ schurPol' 3 [3, 1, 1]
M[3, 1, 1] + M[2, 2, 1]

prettySymmetricQSpray :: QSpray -> String

Prints a symmetric spray as a linear combination of monomial symmetric polynomials

>>> putStrLn $ prettySymmetricQSpray $ jackPol' 3 [3, 1, 1] 2 'J'
42*M[3,1,1] + 28*M[2,2,1]

prettySymmetricQSpray' :: QSpray' -> String

Same as prettySymmetricQSpray but for a QSpray' symmetric spray

prettySymmetricSymbolicQSpray :: String -> SymbolicQSpray -> String

Prints a symmetric symbolic spray as a linear combination of monomial symmetric polynomials

>>> putStrLn $ prettySymmetricSymbolicQSpray "a" $ jackSymbolicPol' 3 [3, 1, 1] 'J'
{ 4*a^2 + 10*a + 6 }*M[3,1,1] + { 8*a + 12 }*M[2,2,1]