Nil-Coxeter Algebra¶

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

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 ARGUMENTS:

• 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] + 2*u[0] + 3*u[1] + 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]
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]
k_schur_noncommutative_variables(la)

In type $$A^{(1)}$$ this is the $$k$$-Schur function in noncommutative variables defined by Thomas Lam [Lam2005].

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]