Nil-Coxeter Algebra

class sage.algebras.nil_coxeter_algebra.NilCoxeterAlgebra(W, base_ring=Rational Field, prefix='u')

Bases: sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra.T

Construct the Nil-Coxeter algebra of given type. This is the algebra with generators u_i for every node i of the corresponding Dynkin diagram. It has the usual braid relations (from the Weyl group) as well as the quadratic relation u_i^2 = 0.

INPUT:

  • W – a Weyl group

OPTIONAL ARGUEMENTS:

  • base_ring – a ring (default is the rational numbers)
  • prefix – a label for the generators (default “u”)

EXAMPLES:

sage: U = NilCoxeterAlgebra(WeylGroup(['A',3,1]))
sage: u0, u1, u2, u3 = U.algebra_generators()
sage: u1*u1
0
sage: u2*u1*u2 == u1*u2*u1
True
sage: U.an_element()
u[0,1,2,3] + 3*u[0,1] + 2*u[0] + 1
homogeneous_generator_noncommutative_variables(r)

Give the r^{th} homogeneous function inside the Nil-Coxeter algebra. In finite type A this is the sum of all decreasing elements of length r. In affine type A this is the sum of all cyclically decreasing elements of length r. This is only defined in finite type A, B and affine types A^{(1)}, B^{(1)}, C^{(1)}, D^{(1)}.

INPUT:

  • r – a positive integer at most the rank of the Weyl group

EXAMPLES:

sage: U = NilCoxeterAlgebra(WeylGroup(['A',3,1]))
sage: U.homogeneous_generator_noncommutative_variables(2)
u[1,0] + u[2,0] + u[0,3] + u[3,2] + u[3,1] + u[2,1]

sage: U = NilCoxeterAlgebra(WeylGroup(['B',4]))
sage: U.homogeneous_generator_noncommutative_variables(2)
u[1,2] + u[2,1] + u[3,1] + u[4,1] + u[2,3] + u[3,2] + u[4,2] + u[3,4] + u[4,3]

sage: U = NilCoxeterAlgebra(WeylGroup(['C',3]))
sage: U.homogeneous_generator_noncommutative_variables(2)
Traceback (most recent call last):
...
AssertionError: Analogue of symmetric functions in noncommutative variables is not defined in type ['C', 3]

TESTS:

sage: U = NilCoxeterAlgebra(WeylGroup(['B',3,1]))
sage: U.homogeneous_generator_noncommutative_variables(-1)
0
sage: U.homogeneous_generator_noncommutative_variables(0)
1
homogeneous_noncommutative_variables(la)

Give the homogeneous function indexed by la, viewed inside the Nil-Coxeter algebra. This is only defined in finite type A, B and affine types A^{(1)}, B^{(1)}, C^{(1)}, D^{(1)}.

INPUT:

  • la – a partition with first part bounded by the rank of the Weyl group

EXAMPLES:

sage: U = NilCoxeterAlgebra(WeylGroup(['B',2,1]))
sage: U.homogeneous_noncommutative_variables([2,1])
u[1,2,0] + 2*u[2,1,0] + u[0,2,0] + u[0,2,1] + u[1,2,1] + u[2,1,2] + u[2,0,2] + u[1,0,2]

TESTS:

sage: U = NilCoxeterAlgebra(WeylGroup(['B',2,1]))
sage: U.homogeneous_noncommutative_variables([])
1
k_schur_noncommutative_variables(la)

In type A^{(1)} this is the k-Schur function in noncommutative variables defined by Thomas Lam.

REFERENCES:

[Lam2005]
  1. Lam, Affine Stanley symmetric functions, Amer. J. Math. 128 (2006), no. 6, 1553–1586.

This function is currently only defined in type A^{(1)}.

INPUT:

  • la – a partition with first part bounded by the rank of the Weyl group

EXAMPLES:

sage: A = NilCoxeterAlgebra(WeylGroup(['A',3,1]))
sage: A.k_schur_noncommutative_variables([2,2])
u[0,3,1,0] + u[3,1,2,0] + u[1,2,0,1] + u[3,2,0,3] + u[2,0,3,1] + u[2,3,1,2]

TESTS:

sage: A = NilCoxeterAlgebra(WeylGroup(['A',3,1]))
sage: A.k_schur_noncommutative_variables([])
1

sage: A.k_schur_noncommutative_variables([1,2])
Traceback (most recent call last):
...
AssertionError: [1, 2] is not a partition.

sage: A.k_schur_noncommutative_variables([4,2])
Traceback (most recent call last):
...
AssertionError: [4, 2] is not a 3-bounded partition.

sage: C = NilCoxeterAlgebra(WeylGroup(['C',3,1]))
sage: C.k_schur_noncommutative_variables([2,2])
Traceback (most recent call last):
...
AssertionError: Weyl Group of type ['C', 3, 1] (as a matrix group acting on the root space) is not affine type A.

Previous topic

Iwahori-Hecke Algebras

Next topic

Affine nilTemperley Lieb Algebra of type A

This Page