# 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 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] 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.


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