Yokonuma-Hecke Algebras

AUTHORS:

  • Travis Scrimshaw (2015-11): initial version
sage.algebras.yokonuma_hecke_algebra.YokonumaHeckeAlgebra

The Yokonuma-Hecke algebra \(Y_{d,n}(q)\).

Let \(R\) be a commutative ring and \(q\) be a unit in \(R\). The Yokonuma-Hecke algebra \(Y_{d,n}(q)\) is the associative, unital \(R\)-algebra generated by \(t_1, t_2, \ldots, t_n, g_1, g_2, \ldots, g_{n-1}\) and subject to the relations:

  • \(g_i g_j = g_j g_i\) for all \(|i - j| > 1\),
  • \(g_i g_{i+1} g_i = g_{i+1} g_i g_{i+1}\),
  • \(t_i t_j = t_j t_i\),
  • \(t_j g_i = g_i t_{j s_i}\), and
  • \(t_j^d = 1\),

where \(s_i\) is the simple transposition \((i, i+1)\), along with the quadratic relation

\[g_i^2 = 1 + \frac{(q - q^{-1})}{d} \left( \sum_{s=0}^{d-1} t_i^s t_{i+1}^{-s} \right) g_i.\]

Thus the Yokonuma-Hecke algebra can be considered a quotient of the framed braid group \((\ZZ / d\ZZ) \wr B_n\), where \(B_n\) is the classical braid group on \(n\) strands, by the quadratic relations. Moreover, all of the algebra generators are invertible. In particular, we have

\[g_i^{-1} = g_i - (q - q^{-1}) e_i.\]

When we specialize \(q = \pm 1\), we obtain the group algebra of the complex reflection group \(G(d, 1, n) = (\ZZ / d\ZZ) \wr S_n\). Moreover for \(d = 1\), the Yokonuma-Hecke algebra is equal to the Iwahori-Hecke of type \(A_{n-1}\).

INPUT:

  • d – the maximum power of \(t\)
  • n – the number of generators
  • q – (optional) an invertible element in a commutative ring; the default is \(q \in \QQ[q,q^{-1}]\)
  • R – (optional) a commutative ring containing q; the default is the parent of \(q\)

EXAMPLES:

We construct \(Y_{4,3}\) and do some computations:

sage: Y = algebras.YokonumaHecke(4, 3)
sage: g1, g2, t1, t2, t3 = Y.algebra_generators()
sage: g1 * g2
g[1,2]
sage: t1 * g1
t1*g[1]
sage: g2 * t2
t3*g[2]
sage: g2 * t3
t2*g[2]
sage: (g2 + t1) * (g1 + t2*t3)
g[2,1] + t2*t3*g[2] + t1*g[1] + t1*t2*t3
sage: g1 * g1
1 - (1/4*q^-1-1/4*q)*g[1] - (1/4*q^-1-1/4*q)*t1*t2^3*g[1]
 - (1/4*q^-1-1/4*q)*t1^2*t2^2*g[1] - (1/4*q^-1-1/4*q)*t1^3*t2*g[1]
sage: g2 * g1 * t1
t3*g[2,1]

We construct the elements \(e_i\) and show that they are idempotents:

sage: e1 = Y.e(1); e1
1/4 + 1/4*t1*t2^3 + 1/4*t1^2*t2^2 + 1/4*t1^3*t2
sage: e1 * e1 == e1
True
sage: e2 = Y.e(2); e2
1/4 + 1/4*t2*t3^3 + 1/4*t2^2*t3^2 + 1/4*t2^3*t3
sage: e2 * e2 == e2
True

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