Bases: sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebraT
Construct the Nil-Coxeter algebra of given type. This is the algebra
with generators for every node
of the corresponding Dynkin
diagram. It has the usual braid relations (from the Weyl group) as well
as the quadratic relation
.
INPUT:
OPTIONAL ARGUEMENTS:
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()
2*u0 + 3*u0*u1 + 1 + u0*u1*u2*u3
Give the homogeneous function inside the Nil-Coxeter algebra.
In finite type
this is the sum of all decreasing elements of length
.
In affine type
this is the sum of all cyclically decreasing elements of length
.
This is only defined in finite type
,
and affine types
,
,
,
.
INPUT:
EXAMPLES:
sage: U = NilCoxeterAlgebra(WeylGroup(['A',3,1]))
sage: U.homogeneous_generator_noncommutative_variables(2)
u1*u0 + u2*u0 + u0*u3 + u3*u2 + u3*u1 + u2*u1
sage: U = NilCoxeterAlgebra(WeylGroup(['B',4]))
sage: U.homogeneous_generator_noncommutative_variables(2)
u1*u2 + u2*u1 + u3*u1 + u4*u1 + u2*u3 + u3*u2 + u4*u2 + u3*u4 + u4*u3
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
Give the homogeneous function indexed by , viewed inside the Nil-Coxeter algebra.
This is only defined in finite type
,
and affine types
,
,
,
.
INPUT:
EXAMPLES:
sage: U = NilCoxeterAlgebra(WeylGroup(['B',2,1]))
sage: U.homogeneous_noncommutative_variables([2,1])
u1*u2*u0 + 2*u2*u1*u0 + u0*u2*u0 + u0*u2*u1 + u1*u2*u1 + u2*u1*u2 + u2*u0*u2 + u1*u0*u2
TESTS:
sage: U = NilCoxeterAlgebra(WeylGroup(['B',2,1]))
sage: U.homogeneous_noncommutative_variables([])
1
In type this is the
-Schur function in noncommutative variables defined by Thomas Lam.
REFERENCES:
[Lam2005]
- Lam, Affine Stanley symmetric functions, Amer. J. Math. 128 (2006), no. 6, 1553–1586.
This function is currently only defined in type .
INPUT:
EXAMPLES:
sage: A = NilCoxeterAlgebra(WeylGroup(['A',3,1]))
sage: A.k_schur_noncommutative_variables([2,2])
u0*u3*u1*u0 + u3*u1*u2*u0 + u1*u2*u0*u1 + u3*u2*u0*u3 + u2*u0*u3*u1 + u2*u3*u1*u2
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.