## 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
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

# 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: 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]

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

* 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```

combinat/MultivariatePolynomials (last edited 2011-05-24 18:14:06 by KelvinLi)