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