Implement mapping symmetric polynomial to Laurent polynomial

asked 2015-07-28 00:01:08 +0100

How can you implement transforming a symmetric polynomial into a Laurent polynomial by mapping some variables to the inverses of others? In other words, given, say, a polynomial in $x_0,y_0,x_1,y_1$, how can we output this polynomial under the map sending $y_i$ to $x_i^{-1}$? I'm specifically looking to apply this to Hall-Littlewood polynomials if that helps.

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answered 2015-07-28 08:29:19 +0100

nbruin
4143 ●3 ●45 ●90

Is the fact that you're interested in symmetric polynomials relevant? For polynomial rings in general defining the homomorphism is quite straightforward, so I would expect that by restricting the map you'll also get it for symmetric polynomials:

sage: R.<x0,x1,y0,y1>=QQ[]
sage: S.<X0,X1>=LaurentPolynomialRing(QQ)
sage: H=Hom(R,S)
sage: m=H([X0,X1,X0^(-1),X1^(-1)])
sage: m(x0+2*x1+3*y0+4*y1)
X0 + 2*X1 + 4*X1^-1 + 3*X0^-1

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That's exactly what I needed, thanks!

rogervanpeski ( 2015-07-28 21:27:52 +0100 )edit

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Asked: 2015-07-28 00:01:08 +0100

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Last updated: Jul 28 '15

Implement mapping symmetric polynomial to Laurent polynomial edit