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sage: m = P.m | sage: x = P.monomial_basis() |
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sage: m.basis().keys() | sage: x.basis().keys() |
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sage: x0,x1,x2 = m.gens() | sage: x0,x1,x2 = x.gens() |
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sage: m.term([3,1,2]) + x2^3 + 3 | sage: x.term([3,1,2]) + x2^3 + 3 |
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sage:: m is MultivariatePolynomialRing(F, 'x0,x1,x2') | sage:: x is MultivariatePolynomialRing(F, 'x0,x1,x2') |
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sage: m.print_style(style = "vectors") sage: m.term([3,1,2]) + x2^3 + 3 |
sage: x.print_style(style = "vectors") sage: x.term([3,1,2]) + x2^3 + 3 |
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sage: Y[1,0,0] * Y[0,1,0] | sage: Y[1,0,0] * Y[0,1,0] # ToY(Y[1,0,0] * Y[0,1,0]) |
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sage: m(Y[1,0,0]) | sage: m(Y[1,0,0]) # Tox(Y[1,0,0]) |
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sage: K(x0 * (x0+x1)) | sage: K(x0 * (x0+x1)) # x2K(x0 * (x0+x1)) En type B (not yet implemented): x2KB(x0 * (x0+x1)) |
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sage: K[1,0,0] * K[0,1,0] | sage: K[1,0,0] * K[0,1,0] # x2K(K2x(K[1,0,0] * K[0,1,0])) |
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sage: m(K[0,2,0] - K[2,0,0]) # K2x(K[0,2,0] - K[2,0,0]) x0^2 + x0*x1 # TODO: add larger examples computed with ACE! The Demazure atoms: / dual of key polynomials: sage: hK = P.key_polynomials_dual(type = "A") # tool(`Key7.mpl`) sage: hK Multivariate polynomials in the dual basis of key polynomials sage: hK.dual() Multivariate polynomials in the key polynomials basis sage: scalar(hK[5,2,4], K[4,2,5]) # Watch for the reversal of the vector (weight) 1 sage: hK.basis().keys() Integer vectors of length 3 sage: hK[2,4,1] * hK[2,1,3] # x2hK(expand(K2x(hK[0,2,1]) * K2x(hK[1,0,1])))) hK[1,2,2] + hK[2,1,2] sage: x(hK[4,0,3]) # K2x(hK[4,0,3]) x[4,2,1] + x[4,1,2] + x[4,0,3] Grothendieck polynomials:: sage: G = P.grothendieck_polynomials(type = "A") sage: G Multivariate polynomials in the key polynomial basis sage: G.basis().keys() Integer vectors of length 3 sage: G(x0 * (x0+x1)) G[0,2,0] - G[2,0,0] sage: G[1,0,0] * G[0,1,0] G[1,1,0] + G[2,0,0] |
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The Demazure atoms:: sage: HK = P.demazure_atoms(type = "A") Or possibly:: sage: HK = K.dual() sage: HK Multivariate polynomials in the demazure atoms basis sage: HK.dual() Multivariate polynomials in the key polynomial basis sage: scalar(HK[5,2,4], K[4,2,5]) # Watch for the reversal of the vector (weight) 1 sage: HK.basis().keys() Integer vectors of length 3 sage: H sage: HK[1,0,0] * HK[0,1,0] K[1,1,0] + K[2,0,0] sage: m(Y[1,0,0]) x0 # TODO: add larger examples! |
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Specifications for the abstract ring of multivariate polynomials, with several bases
Ticket 6629
First micro draft
Setup the framework for MultivariatePolynomials with several bases: Let us work over `F=\QQ(q,t)` (will be needed for Macdonald polynomials):: sage: F = FractionField(QQ['q,t']); (q,t) = F.gens(); F.rename('QQ(q,t)') We construct an (abstract) ring of multivariate polynomials over F:: sage: P = AbstractPolynomialRing(F, 'x0,x1,x2'); P The abstract ring of multivariate polynomials in x0, x1, x2 over QQ(q,t) See [[http://mupad-combinat.svn.sourceforge.net/viewvc/mupad-combinat/trunk/MuPAD-Combinat/lib/EXAMPLES/MultivariatePolynomials.mu?view=markup|examples::MultivariatePolynomials]] ([[http://mupad-combinat.svn.sourceforge.net/viewvc/mupad-combinat/trunk/MuPAD-Combinat/lib/EXAMPLES/TEST/MultivariatePolynomials.tst?view=markup|Tests]] for a preliminary implementation in MuPAD-Combinat This ring has several bases, starting with the usual monomial basis:: sage: x = P.monomial_basis() Multivariate Polynomial Ring in x0, x1, x2 over QQ(q,t) sage: x.basis().keys() Integer vectors of length 3 sage: x0,x1,x2 = x.gens() sage: x.term([3,1,2]) + x2^3 + 3 x0^3*x1*x2^2 + x2^3 + 3 sage:: x is MultivariatePolynomialRing(F, 'x0,x1,x2') True sage: x.print_style(style = "vectors") sage: x.term([3,1,2]) + x2^3 + 3 x[3,1,2] + x[0,0,3] + 3*x[0,0,0] The Schubert basis:: sage: Y = P.schubert() sage: Y Multivariate polynomials in the Schubert basis sage: Y.basis().keys() Integer vectors of length 3 sage: Y[1,0,0] * Y[0,1,0] # ToY(Y[1,0,0] * Y[0,1,0]) Y[1,1,0] + Y[2,0,0] sage: m(Y[1,0,0]) # Tox(Y[1,0,0]) x0 # TODO: add larger examples! # Design to be discussed in the long run # One can optionaly specify an alphabet, whose elements should leave in the ground ring:: # sage: Y = P.schubert([x2,x1,x0]) # If the ground ring allows for it, the default alphabet should be [y0,y1,y2] The key polynomials (Demazure characters):: sage: K = P.key_polynomials(type = "A") sage: K Multivariate polynomials in the key polynomial basis sage: K.basis().keys() Integer vectors of length 3 sage: K(x0 * (x0+x1)) # x2K(x0 * (x0+x1)) En type B (not yet implemented): x2KB(x0 * (x0+x1)) K[0,2,0] - K[2,0,0] sage: K[1,0,0] * K[0,1,0] # x2K(K2x(K[1,0,0] * K[0,1,0])) K[1,1,0] + K[2,0,0] sage: m(K[0,2,0] - K[2,0,0]) # K2x(K[0,2,0] - K[2,0,0]) x0^2 + x0*x1 # TODO: add larger examples computed with ACE! The Demazure atoms: / dual of key polynomials: sage: hK = P.key_polynomials_dual(type = "A") # tool(`Key7.mpl`) sage: hK Multivariate polynomials in the dual basis of key polynomials sage: hK.dual() Multivariate polynomials in the key polynomials basis sage: scalar(hK[5,2,4], K[4,2,5]) # Watch for the reversal of the vector (weight) 1 sage: hK.basis().keys() Integer vectors of length 3 sage: hK[2,4,1] * hK[2,1,3] # x2hK(expand(K2x(hK[0,2,1]) * K2x(hK[1,0,1])))) hK[1,2,2] + hK[2,1,2] sage: x(hK[4,0,3]) # K2x(hK[4,0,3]) x[4,2,1] + x[4,1,2] + x[4,0,3] Grothendieck polynomials:: sage: G = P.grothendieck_polynomials(type = "A") sage: G Multivariate polynomials in the key polynomial basis sage: G.basis().keys() Integer vectors of length 3 sage: G(x0 * (x0+x1)) G[0,2,0] - G[2,0,0] sage: G[1,0,0] * G[0,1,0] G[1,1,0] + G[2,0,0] sage: m(K[0,2,0] - K[2,0,0]) x0^2 + x0*x1 # TODO: add larger examples computed with ACE! The SchurSchubert basis (see MuPAD):: sage:: P.SchurSchubert() This is the free module over Schur polynomials with basis Schubert polynomials; the later are indexed by (the code of) permutations of `S_n`. sage:: P.coeffRing() Symmetric polynomials in the Schur basis over QQ(q,t) sage:: P.basis().keys() Permutations of S_n ? or Codes ? sage:: P.basis().cardinality() 6 Other bases in MuPAD-Combinat: * NonSymmetricHL, NonSymmetricHLdual * UniversalDecompositionAlgebra (free module over symmetric functions in the e basis, with monomial below the stair as basis * FreeModule over symmetric functions in the e basis over t, with descent monomials as basis. Non symmetric Macdonald polynomials (should recycle the current sage.combinat.sf.ns_macdonald) sage: Macdo = P.MacdonaldPolynomials(q, t) sage: E = Macdo.E(pi = [3,1,2]); E Multivariate Polynomial Ring in x0, x1, x2 over QQ(q,t), in the Macdonald E basis, with basement [3,1,2] sage: E[1,0,0] E[1,0,0] sage: m(E[1,0,0]) x0