Ariki-Koike Algebras#

The Ariki-Koike algebras were introduced by Ariki and Koike [AK1994] as a natural generalization of the Iwahori-Hecke algebras of types $A$ and $B$ (see IwahoriHeckeAlgebra). Soon afterwards, Broué and Malle defined analogues of the Hecke algebras for all complex reflection groups

Fix non-negative integers $r$ an $n$ . The Ariki-Koike algebras are deformations of the group algebra of the complex reflection group $G (r, 1, n) = Z / r Z ≀ S_{n}$ . If $R$ is a ring containing a Hecke parameter $q$ and cyclotomic parameters $u_{0}, \dots, u_{r - 1}$ then the Ariki-Koike algebra $H_{n} (q, u_{1}, \dots, u_{r})$ is the unital associative $r$ -algebra with generators $T_{0}, T_{1}, \dots, T_{n - 1}$ an relations:

\begin{array}{r} \begin{aligned} \prod_{i = 0}^{r - 1} (T_{0} - u_{i}) & = 0, \\ T_{i}^{2} & = (q - 1) T_{i} + q & for 1 \leq i < n, \\ T_{0} T_{1} T_{0} T_{1} & = T_{1} T_{0} T_{1} T_{0}, \\ T_{i} T_{j} & = T_{j} T_{i} & if | i - j | \geq 2, \\ T_{i} T_{i + 1} T_{i} & = T_{i + 1} T_{i} T_{i + 1} & for 1 \leq i < n . \end{aligned} \end{array}

AUTHORS:

Travis Scrimshaw (2016-04): initial version
Andrew Mathas (2016-07): improved multiplication code

REFERENCES:

class sage.algebras.hecke_algebras.ariki_koike_algebra.ArikiKoikeAlgebra(r, n, q, u, R)[source]#

Bases: Parent, UniqueRepresentation

The Ariki-Koike algebra $H_{r, n} (q, u)$ .

Let $R$ be an unital integral domain. Let $q, u_{0}, \dots, u_{r - 1} \in R$ such that $q^{- 1} \in R$ . The Ariki-Koike algebra is the unital associative algebra $H_{r, n} (q, u)$ generated by $T_{0}, \dots, T_{n - 1}$ that satisfies the following relations:

\begin{array}{r} \begin{aligned} \prod_{i = 0}^{r - 1} (T_{0} - u_{i}) & = 0, \\ T_{i}^{2} & = (q - 1) T_{i} + q & for 1 \leq i < n, \\ T_{0} T_{1} T_{0} T_{1} & = T_{1} T_{0} T_{1} T_{0}, \\ T_{i} T_{j} & = T_{j} T_{i} & if | i - j | \geq 2, \\ T_{i} T_{i + 1} T_{i} & = T_{i + 1} T_{i} T_{i + 1} & for 1 \leq i < n . \end{aligned} \end{array}

The parameter $q$ is called the Hecke parameter and the parameters $u_{0}, \dots, u_{r - 1}$ are called the cyclotomic parameters. Thus, the Ariki-Koike algebra is a deformation of the group algebra of the complex reflection group $G (r, 1, n) = Z / r Z ≀ S_{n}$ .

Next, we define Jucys-Murphy elements

L_{i} = q^{- i + 1} T_{i - 1} \dots T_{1} T_{0} T_{1} \dots T_{i - 1}

for $1 \leq i \leq n$ .

Note

These element differ by a power of $q$ from the corresponding elements in [AK1994]. However, these elements are more commonly used because they lead to nicer representation theoretic formulas.

Ariki and Koike [AK1994] showed that $H_{r, n} (q, u)$ is a free $R$ -module with a basis given by

{L_{1}^{c_{i}} \dots L_{n}^{c_{n}} T_{w} ∣ w \in S_{n}, 0 \leq c_{i} < r} .

In particular, we have $\dim H_{r, n} (q, u) = r^{n} n! = | G (r, 1, n) |$ . Moreover, we have $L_{i} L_{j} = L_{i} L_{j}$ for all $1 \leq i, j \leq n$ .

The Ariki-Koike algebra $H_{r, n} (q, u)$ can be considered as a quotient of the group algebra of the braid group for $G (r, 1, n)$ by the ideal generated by $\prod_{i = 0}^{r - 1} (T_{0} - u_{i})$ and $(T_{i} - q) (T_{i} + 1)$ . Furthermore, $H_{r, n} (q, u)$ can be constructed as a quotient of the extended affine Hecke algebra of type $A_{n - 1}^{(1)}$ by $\prod_{i = 0}^{r - 1} (X_{1} - u_{i})$ .

Since the Ariki-Koike algebra is a quotient of the group algebra of the braid group of $G (r, 1, n)$ , we can recover the group algebra of $G (r, 1, n)$ as follows. Consider $u = (1, ζ_{r}, \dots, ζ_{r}^{r - 1})$ , where $ζ_{r}$ is a primitive $r$ -th root of unity, then we have

R G (r, 1, n) = H_{r, n} (1, u) .

INPUT:

r – the maximum power of $L_{i}$
n – the rank $S_{n}$
q – (optional) an invertible element in a commutative ring; the default is $q \in R [q, q^{- 1}]$ , where $R$ is the ring containing the variables u
u – (optional) the variables $u_{1}, \dots, u_{r}$ ; the default is the generators of $Z [u_{1}, \dots, u_{r}]$
R – (optional) a commutative ring containing q and u; the default is the parent of $q$ and $u_{1}, \dots, u_{r}$
use_fraction_field – (default: False) whether to use the fraction field or not

EXAMPLES:

We start by constructing an Ariki-Koike algebra where the values $q, u$ are generic and do some computations:

Sage

sage: H = algebras.ArikiKoike(3, 4)

Sage Live

H = algebras.ArikiKoike(3, 4)

Python

>>> from sage.all import *
>>> H = algebras.ArikiKoike(Integer(3), Integer(4))

Next, we do some computations using the $L T$ basis:

Sage

sage: LT = H.LT()
sage: LT.inject_variables()
Defining L1, L2, L3, L4, T1, T2, T3
sage: T1 * T2 * T1 * T2
q*T[2,1] - (1-q)*T[2,1,2]
sage: T1 * L1 * T2 * L3 * T1 * T2
-(q-q^2)*L2*L3*T[2] + q*L1*L2*T[2,1] - (1-q)*L1*L2*T[2,1,2]
sage: L1^3
u0*u1*u2 + ((-u0*u1-u0*u2-u1*u2))*L1 + ((u0+u1+u2))*L1^2
sage: L3 * L2 * L1
L1*L2*L3
sage: u = LT.u()
sage: q = LT.q()
sage: (q + 2*u[0]) * (T1 * T2) * L3
(-2*u0+(2*u0-1)*q+q^2)*L3*T[1] + (-2*u0+(2*u0-1)*q+q^2)*L2*T[2]
 + (2*u0+q)*L1*T[1,2]

Sage Live

LT = H.LT()
LT.inject_variables()
T1 * T2 * T1 * T2
T1 * L1 * T2 * L3 * T1 * T2
L1^3
L3 * L2 * L1
u = LT.u()
q = LT.q()
(q + 2*u[0]) * (T1 * T2) * L3

Python

>>> from sage.all import *
>>> LT = H.LT()
>>> LT.inject_variables()
Defining L1, L2, L3, L4, T1, T2, T3
>>> T1 * T2 * T1 * T2
q*T[2,1] - (1-q)*T[2,1,2]
>>> T1 * L1 * T2 * L3 * T1 * T2
-(q-q^2)*L2*L3*T[2] + q*L1*L2*T[2,1] - (1-q)*L1*L2*T[2,1,2]
>>> L1**Integer(3)
u0*u1*u2 + ((-u0*u1-u0*u2-u1*u2))*L1 + ((u0+u1+u2))*L1^2
>>> L3 * L2 * L1
L1*L2*L3
>>> u = LT.u()
>>> q = LT.q()
>>> (q + Integer(2)*u[Integer(0)]) * (T1 * T2) * L3
(-2*u0+(2*u0-1)*q+q^2)*L3*T[1] + (-2*u0+(2*u0-1)*q+q^2)*L2*T[2]
 + (2*u0+q)*L1*T[1,2]

We check the defining relations:

Sage

sage: prod(L1 - val for val in u) == H.zero()
True
sage: L1 * T1 * L1 * T1 == T1 * L1 * T1 * L1
True
sage: T1 * T2 * T1 == T2 * T1 * T2
True
sage: T2 * T3 * T2 == T3 * T2 * T3
True
sage: L2 == q^-1 * T1 * L1 * T1
True
sage: L3 == q^-2 * T2 * T1 * L1 * T1 * T2
True

Sage Live

prod(L1 - val for val in u) == H.zero()
L1 * T1 * L1 * T1 == T1 * L1 * T1 * L1
T1 * T2 * T1 == T2 * T1 * T2
T2 * T3 * T2 == T3 * T2 * T3
L2 == q^-1 * T1 * L1 * T1
L3 == q^-2 * T2 * T1 * L1 * T1 * T2

Python

>>> from sage.all import *
>>> prod(L1 - val for val in u) == H.zero()
True
>>> L1 * T1 * L1 * T1 == T1 * L1 * T1 * L1
True
>>> T1 * T2 * T1 == T2 * T1 * T2
True
>>> T2 * T3 * T2 == T3 * T2 * T3
True
>>> L2 == q**-Integer(1) * T1 * L1 * T1
True
>>> L3 == q**-Integer(2) * T2 * T1 * L1 * T1 * T2
True

We construct an Ariki-Koike algebra with $u = (1, ζ_{3}, ζ_{3}^{2})$ , where $ζ_{3}$ is a primitive third root of unity:

Sage

sage: # needs sage.rings.number_field
sage: F = CyclotomicField(3)
sage: zeta3 = F.gen()
sage: R.<q> = LaurentPolynomialRing(F)
sage: H = algebras.ArikiKoike(3, 4, q=q, u=[1, zeta3, zeta3^2], R=R)
sage: H.LT().inject_variables()
Defining L1, L2, L3, L4, T1, T2, T3
sage: L1^3
1
sage: L2^3
1 - (q^-1-1)*T[1] - (q^-1-1)*L1*L2^2*T[1] - (q^-1-1)*L1^2*L2*T[1]

Sage Live

# needs sage.rings.number_field
F = CyclotomicField(3)
zeta3 = F.gen()
R.<q> = LaurentPolynomialRing(F)
H = algebras.ArikiKoike(3, 4, q=q, u=[1, zeta3, zeta3^2], R=R)
H.LT().inject_variables()
L1^3
L2^3

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> F = CyclotomicField(Integer(3))
>>> zeta3 = F.gen()
>>> R = LaurentPolynomialRing(F, names=('q',)); (q,) = R._first_ngens(1)
>>> H = algebras.ArikiKoike(Integer(3), Integer(4), q=q, u=[Integer(1), zeta3, zeta3**Integer(2)], R=R)
>>> H.LT().inject_variables()
Defining L1, L2, L3, L4, T1, T2, T3
>>> L1**Integer(3)
1
>>> L2**Integer(3)
1 - (q^-1-1)*T[1] - (q^-1-1)*L1*L2^2*T[1] - (q^-1-1)*L1^2*L2*T[1]

Next, we additionally take $q = 1$ to obtain the group algebra of $G (r, 1, n)$ :

Sage

sage: # needs sage.rings.number_field
sage: F = CyclotomicField(3)
sage: zeta3 = F.gen()
sage: H = algebras.ArikiKoike(3, 4, q=1, u=[1, zeta3, zeta3^2], R=F)
sage: LT = H.LT()
sage: LT.inject_variables()
Defining L1, L2, L3, L4, T1, T2, T3
sage: A = ColoredPermutations(3, 4).algebra(F)
sage: s1, s2, s3, s0 = list(A.algebra_generators())
sage: all(L^3 == LT.one() for L in LT.L())
True
sage: J = [s0, s3*s0*s3, s2*s3*s0*s3*s2, s1*s2*s3*s0*s3*s2*s1]
sage: all(Ji^3 == A.one() for Ji in J)
True

Sage Live

# needs sage.rings.number_field
F = CyclotomicField(3)
zeta3 = F.gen()
H = algebras.ArikiKoike(3, 4, q=1, u=[1, zeta3, zeta3^2], R=F)
LT = H.LT()
LT.inject_variables()
A = ColoredPermutations(3, 4).algebra(F)
s1, s2, s3, s0 = list(A.algebra_generators())
all(L^3 == LT.one() for L in LT.L())
J = [s0, s3*s0*s3, s2*s3*s0*s3*s2, s1*s2*s3*s0*s3*s2*s1]
all(Ji^3 == A.one() for Ji in J)

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> F = CyclotomicField(Integer(3))
>>> zeta3 = F.gen()
>>> H = algebras.ArikiKoike(Integer(3), Integer(4), q=Integer(1), u=[Integer(1), zeta3, zeta3**Integer(2)], R=F)
>>> LT = H.LT()
>>> LT.inject_variables()
Defining L1, L2, L3, L4, T1, T2, T3
>>> A = ColoredPermutations(Integer(3), Integer(4)).algebra(F)
>>> s1, s2, s3, s0 = list(A.algebra_generators())
>>> all(L**Integer(3) == LT.one() for L in LT.L())
True
>>> J = [s0, s3*s0*s3, s2*s3*s0*s3*s2, s1*s2*s3*s0*s3*s2*s1]
>>> all(Ji**Integer(3) == A.one() for Ji in J)
True

class LT(algebra)[source]#

Bases: _Basis

The basis of the Ariki-Koike algebra given by monomials of the form $L T$ , where $L$ is product of Jucys-Murphy elements and $T$ is a product of ${T_{i} | 0 < i < n}$ .

This was the basis defined in [AK1994] except using the renormalized Jucys-Murphy elements.

class Element[source]#: Bases: IndexedFreeModuleElement

L(i=None)[source]#

Return the generator(s) $L_{i}$ .

INPUT:

i – (default: None) the generator $L_{i}$ or if None, then the list of all generators $L_{i}$

EXAMPLES:

Sage

sage: LT = algebras.ArikiKoike(8, 3).LT()
sage: LT.L(2)
L2
sage: LT.L()
[L1, L2, L3]

sage: LT = algebras.ArikiKoike(1, 3).LT()
sage: LT.L(2)
u + (-u*q^-1+u)*T[1]
sage: LT.L()
[u,
 u + (-u*q^-1+u)*T[1],
 u + (-u*q^-1+u)*T[2] + (-u*q^-2+u*q^-1)*T[2,1,2]]

Sage Live

LT = algebras.ArikiKoike(8, 3).LT()
LT.L(2)
LT.L()
LT = algebras.ArikiKoike(1, 3).LT()
LT.L(2)
LT.L()

Python

>>> from sage.all import *
>>> LT = algebras.ArikiKoike(Integer(8), Integer(3)).LT()
>>> LT.L(Integer(2))
L2
>>> LT.L()
[L1, L2, L3]

>>> LT = algebras.ArikiKoike(Integer(1), Integer(3)).LT()
>>> LT.L(Integer(2))
u + (-u*q^-1+u)*T[1]
>>> LT.L()
[u,
 u + (-u*q^-1+u)*T[1],
 u + (-u*q^-1+u)*T[2] + (-u*q^-2+u*q^-1)*T[2,1,2]]

T(i=None)[source]#

Return the generator(s) $T_{i}$ of self.

INPUT:

i – (default: None) the generator $T_{i}$ or if None, then the list of all generators $T_{i}$

EXAMPLES:

Sage

sage: LT = algebras.ArikiKoike(8, 3).LT()
sage: LT.T(1)
T[1]
sage: LT.T()
[L1, T[1], T[2]]
sage: LT.T(0)
L1

Sage Live

LT = algebras.ArikiKoike(8, 3).LT()
LT.T(1)
LT.T()
LT.T(0)

Python

>>> from sage.all import *
>>> LT = algebras.ArikiKoike(Integer(8), Integer(3)).LT()
>>> LT.T(Integer(1))
T[1]
>>> LT.T()
[L1, T[1], T[2]]
>>> LT.T(Integer(0))
L1

algebra_generators()[source]#

Return the algebra generators of self.

EXAMPLES:

Sage

sage: LT = algebras.ArikiKoike(5, 3).LT()
sage: dict(LT.algebra_generators())
{'L1': L1, 'L2': L2, 'L3': L3, 'T1': T[1], 'T2': T[2]}

sage: LT = algebras.ArikiKoike(1, 4).LT()
sage: dict(LT.algebra_generators())
{'T1': T[1], 'T2': T[2], 'T3': T[3]}

Sage Live

LT = algebras.ArikiKoike(5, 3).LT()
dict(LT.algebra_generators())
LT = algebras.ArikiKoike(1, 4).LT()
dict(LT.algebra_generators())

Python

>>> from sage.all import *
>>> LT = algebras.ArikiKoike(Integer(5), Integer(3)).LT()
>>> dict(LT.algebra_generators())
{'L1': L1, 'L2': L2, 'L3': L3, 'T1': T[1], 'T2': T[2]}

>>> LT = algebras.ArikiKoike(Integer(1), Integer(4)).LT()
>>> dict(LT.algebra_generators())
{'T1': T[1], 'T2': T[2], 'T3': T[3]}

inverse_T(i)[source]#

Return the inverse of the generator $T_{i}$ .

From the quadratic relation, we have

T_{i}^{- 1} = q^{- 1} T_{i} + (q^{- 1} - 1) .

EXAMPLES:

Sage

sage: LT = algebras.ArikiKoike(3, 4).LT()
sage: [LT.inverse_T(i) for i in range(1, 4)]
[(q^-1-1) + (q^-1)*T[1],
 (q^-1-1) + (q^-1)*T[2],
 (q^-1-1) + (q^-1)*T[3]]

Sage Live

LT = algebras.ArikiKoike(3, 4).LT()
[LT.inverse_T(i) for i in range(1, 4)]

Python

>>> from sage.all import *
>>> LT = algebras.ArikiKoike(Integer(3), Integer(4)).LT()
>>> [LT.inverse_T(i) for i in range(Integer(1), Integer(4))]
[(q^-1-1) + (q^-1)*T[1],
 (q^-1-1) + (q^-1)*T[2],
 (q^-1-1) + (q^-1)*T[3]]

product_on_basis(m1, m2)[source]#

Return the product of the basis elements indexed by m1 and m2.

EXAMPLES:

Sage

sage: LT = algebras.ArikiKoike(6, 3).LT()
sage: m = ((1, 0, 2), Permutations(3)([2,1,3]))
sage: LT.product_on_basis(m, m)
q*L1*L2*L3^4

sage: LT = algebras.ArikiKoike(4, 3).LT()
sage: L1,L2,L3,T1,T2 = LT.algebra_generators()
sage: L1 * T1 * L1^2 * T1
q*L1*L2^2 + (1-q)*L1^2*L2*T[1]
sage: L1^2 * T1 * L1^2 * T1
q*L1^2*L2^2 + (1-q)*L1^3*L2*T[1]
sage: L1^3 * T1 * L1^2 * T1
(-u0*u1*u2*u3+u0*u1*u2*u3*q)*L2*T[1]
 + ((u0*u1*u2+u0*u1*u3+u0*u2*u3+u1*u2*u3)+(-u0*u1*u2-u0*u1*u3-u0*u2*u3-u1*u2*u3)*q)*L1*L2*T[1]
 + ((-u0*u1-u0*u2-u1*u2-u0*u3-u1*u3-u2*u3)+(u0*u1+u0*u2+u1*u2+u0*u3+u1*u3+u2*u3)*q)*L1^2*L2*T[1]
 + ((u0+u1+u2+u3)+(-u0-u1-u2-u3)*q)*L1^3*L2*T[1] + q*L1^3*L2^2

sage: L1^2 * T1 * L1^3 * T1
(-u0*u1*u2*u3+u0*u1*u2*u3*q)*L2*T[1]
 + ((u0*u1*u2+u0*u1*u3+u0*u2*u3+u1*u2*u3)+(-u0*u1*u2-u0*u1*u3-u0*u2*u3-u1*u2*u3)*q)*L1*L2*T[1]
 + ((-u0*u1-u0*u2-u1*u2-u0*u3-u1*u3-u2*u3)+(u0*u1+u0*u2+u1*u2+u0*u3+u1*u3+u2*u3)*q)*L1^2*L2*T[1]
 + q*L1^2*L2^3
 + ((u0+u1+u2+u3)+(-u0-u1-u2-u3)*q)*L1^3*L2*T[1]
 + (1-q)*L1^3*L2^2*T[1]

sage: L1^2 * T1*T2*T1 * L2 * L3 * T2
(q-2*q^2+q^3)*L1^2*L2*L3 - (1-2*q+2*q^2-q^3)*L1^2*L2*L3*T[2]
 - (q-q^2)*L1^3*L3*T[1] + (1-2*q+q^2)*L1^3*L3*T[1,2]
 + q*L1^3*L2*T[2,1] - (1-q)*L1^3*L2*T[2,1,2]

sage: LT = algebras.ArikiKoike(2, 3).LT()
sage: L3 = LT.L(3)
sage: x = LT.an_element()
sage: (x * L3) * L3 == x * (L3 * L3)
True

Sage Live

LT = algebras.ArikiKoike(6, 3).LT()
m = ((1, 0, 2), Permutations(3)([2,1,3]))
LT.product_on_basis(m, m)
LT = algebras.ArikiKoike(4, 3).LT()
L1,L2,L3,T1,T2 = LT.algebra_generators()
L1 * T1 * L1^2 * T1
L1^2 * T1 * L1^2 * T1
L1^3 * T1 * L1^2 * T1
L1^2 * T1 * L1^3 * T1
L1^2 * T1*T2*T1 * L2 * L3 * T2
LT = algebras.ArikiKoike(2, 3).LT()
L3 = LT.L(3)
x = LT.an_element()
(x * L3) * L3 == x * (L3 * L3)

Python

>>> from sage.all import *
>>> LT = algebras.ArikiKoike(Integer(6), Integer(3)).LT()
>>> m = ((Integer(1), Integer(0), Integer(2)), Permutations(Integer(3))([Integer(2),Integer(1),Integer(3)]))
>>> LT.product_on_basis(m, m)
q*L1*L2*L3^4

>>> LT = algebras.ArikiKoike(Integer(4), Integer(3)).LT()
>>> L1,L2,L3,T1,T2 = LT.algebra_generators()
>>> L1 * T1 * L1**Integer(2) * T1
q*L1*L2^2 + (1-q)*L1^2*L2*T[1]
>>> L1**Integer(2) * T1 * L1**Integer(2) * T1
q*L1^2*L2^2 + (1-q)*L1^3*L2*T[1]
>>> L1**Integer(3) * T1 * L1**Integer(2) * T1
(-u0*u1*u2*u3+u0*u1*u2*u3*q)*L2*T[1]
 + ((u0*u1*u2+u0*u1*u3+u0*u2*u3+u1*u2*u3)+(-u0*u1*u2-u0*u1*u3-u0*u2*u3-u1*u2*u3)*q)*L1*L2*T[1]
 + ((-u0*u1-u0*u2-u1*u2-u0*u3-u1*u3-u2*u3)+(u0*u1+u0*u2+u1*u2+u0*u3+u1*u3+u2*u3)*q)*L1^2*L2*T[1]
 + ((u0+u1+u2+u3)+(-u0-u1-u2-u3)*q)*L1^3*L2*T[1] + q*L1^3*L2^2

>>> L1**Integer(2) * T1 * L1**Integer(3) * T1
(-u0*u1*u2*u3+u0*u1*u2*u3*q)*L2*T[1]
 + ((u0*u1*u2+u0*u1*u3+u0*u2*u3+u1*u2*u3)+(-u0*u1*u2-u0*u1*u3-u0*u2*u3-u1*u2*u3)*q)*L1*L2*T[1]
 + ((-u0*u1-u0*u2-u1*u2-u0*u3-u1*u3-u2*u3)+(u0*u1+u0*u2+u1*u2+u0*u3+u1*u3+u2*u3)*q)*L1^2*L2*T[1]
 + q*L1^2*L2^3
 + ((u0+u1+u2+u3)+(-u0-u1-u2-u3)*q)*L1^3*L2*T[1]
 + (1-q)*L1^3*L2^2*T[1]

>>> L1**Integer(2) * T1*T2*T1 * L2 * L3 * T2
(q-2*q^2+q^3)*L1^2*L2*L3 - (1-2*q+2*q^2-q^3)*L1^2*L2*L3*T[2]
 - (q-q^2)*L1^3*L3*T[1] + (1-2*q+q^2)*L1^3*L3*T[1,2]
 + q*L1^3*L2*T[2,1] - (1-q)*L1^3*L2*T[2,1,2]

>>> LT = algebras.ArikiKoike(Integer(2), Integer(3)).LT()
>>> L3 = LT.L(Integer(3))
>>> x = LT.an_element()
>>> (x * L3) * L3 == x * (L3 * L3)
True

class T(algebra)[source]#

Bases: _Basis

The basis of the Ariki-Koike algebra given by monomials of the generators ${T_{i} | 0 \leq i < n}$ .

We use the choice of reduced expression given by [BM1997]:

T_{1, a_{1}} \dots T_{n, a_{n}} T_{w},

where $T_{i, k} = T_{i - 1} \dots T_{2} T_{1} T_{0}^{k}$ (note that $T_{1, k} = T_{0}^{k}$ ) and $w$ is a reduced expression of an element in $S_{n}$ .

L(i=None)[source]#

Return the Jucys-Murphy element(s) $L_{i}$ .

The Jucys-Murphy element $L_{i}$ is defined as

L_{i} = q^{- i + 1} T_{i - 1} \dots T_{1} T_{0} T_{1} \dots T_{i - 1} = q^{- 1} T_{i - 1} L_{i - 1} T_{i - 1} .

INPUT:

i – (default: None) the Jucys-Murphy element $L_{i}$ or if None, then the list of all $L_{i}$

EXAMPLES:

Sage

sage: T = algebras.ArikiKoike(8, 3).T()
sage: T.L(2)
(q^-1)*T[1,0,1]
sage: T.L()
[T[0], (q^-1)*T[1,0,1], (q^-2)*T[2,1,0,1,2]]

sage: T0,T1,T2 = T.T()
sage: q = T.q()
sage: T.L(1) == T0
True
sage: T.L(2) == q^-1 * T1*T0*T1
True
sage: T.L(3) == q^-2 * T2*T1*T0*T1*T2
True

sage: T = algebras.ArikiKoike(1, 3).T()
sage: T.L(2)
u + (-u*q^-1+u)*T[1]
sage: T.L()
[u,
 u + (-u*q^-1+u)*T[1],
 u + (-u*q^-1+u)*T[2] + (-u*q^-2+u*q^-1)*T[2,1,2]]

Sage Live

T = algebras.ArikiKoike(8, 3).T()
T.L(2)
T.L()
T0,T1,T2 = T.T()
q = T.q()
T.L(1) == T0
T.L(2) == q^-1 * T1*T0*T1
T.L(3) == q^-2 * T2*T1*T0*T1*T2
T = algebras.ArikiKoike(1, 3).T()
T.L(2)
T.L()

Python

>>> from sage.all import *
>>> T = algebras.ArikiKoike(Integer(8), Integer(3)).T()
>>> T.L(Integer(2))
(q^-1)*T[1,0,1]
>>> T.L()
[T[0], (q^-1)*T[1,0,1], (q^-2)*T[2,1,0,1,2]]

>>> T0,T1,T2 = T.T()
>>> q = T.q()
>>> T.L(Integer(1)) == T0
True
>>> T.L(Integer(2)) == q**-Integer(1) * T1*T0*T1
True
>>> T.L(Integer(3)) == q**-Integer(2) * T2*T1*T0*T1*T2
True

>>> T = algebras.ArikiKoike(Integer(1), Integer(3)).T()
>>> T.L(Integer(2))
u + (-u*q^-1+u)*T[1]
>>> T.L()
[u,
 u + (-u*q^-1+u)*T[1],
 u + (-u*q^-1+u)*T[2] + (-u*q^-2+u*q^-1)*T[2,1,2]]

T(i=None)[source]#

Return the generator(s) $T_{i}$ of self.

INPUT:

i – (default: None) the generator $T_{i}$ or if None, then the list of all generators $T_{i}$

EXAMPLES:

Sage

sage: T = algebras.ArikiKoike(8, 3).T()
sage: T.T(1)
T[1]
sage: T.T()
[T[0], T[1], T[2]]

sage: T = algebras.ArikiKoike(1, 4).T()

Sage Live

T = algebras.ArikiKoike(8, 3).T()
T.T(1)
T.T()
T = algebras.ArikiKoike(1, 4).T()

Python

>>> from sage.all import *
>>> T = algebras.ArikiKoike(Integer(8), Integer(3)).T()
>>> T.T(Integer(1))
T[1]
>>> T.T()
[T[0], T[1], T[2]]

>>> T = algebras.ArikiKoike(Integer(1), Integer(4)).T()

algebra_generators()[source]#

Return the algebra generators of self.

EXAMPLES:

Sage

sage: T = algebras.ArikiKoike(5, 3).T()
sage: dict(T.algebra_generators())
{0: T[0], 1: T[1], 2: T[2]}

sage: T = algebras.ArikiKoike(1, 4).T()
sage: dict(T.algebra_generators())
{1: T[1], 2: T[2], 3: T[3]}

Sage Live

T = algebras.ArikiKoike(5, 3).T()
dict(T.algebra_generators())
T = algebras.ArikiKoike(1, 4).T()
dict(T.algebra_generators())

Python

>>> from sage.all import *
>>> T = algebras.ArikiKoike(Integer(5), Integer(3)).T()
>>> dict(T.algebra_generators())
{0: T[0], 1: T[1], 2: T[2]}

>>> T = algebras.ArikiKoike(Integer(1), Integer(4)).T()
>>> dict(T.algebra_generators())
{1: T[1], 2: T[2], 3: T[3]}

product_on_basis(m1, m2)[source]#

Return the product of the basis elements indexed by m1 and m2.

EXAMPLES:

Sage

sage: T = algebras.ArikiKoike(2, 3).T()
sage: T0, T1, T2 = T.T()
sage: T.product_on_basis(T0.leading_support(), T1.leading_support())
T[0,1]
sage: T1 * T2
T[1,2]
sage: T2 * T1
T[2,1]
sage: T2 * (T2 * T1 * T0)
-(1-q)*T[2,1,0] + q*T[1,0]
sage: (T1 * T0 * T1 * T0) * T0
(-u0*u1)*T[1,0,1] + ((u0+u1))*T[0,1,0,1]
sage: (T0 * T1 * T0 * T1) * (T0 * T1)
(-u0*u1*q)*T[1,0] + (u0*u1-u0*u1*q)*T[1,0,1]
 + ((u0+u1)*q)*T[0,1,0] + ((-u0-u1)+(u0+u1)*q)*T[0,1,0,1]
sage: T1 * (T0 * T2 * T1 * T0)
T[1,0,2,1,0]
sage: (T1 * T2) * (T2 * T1 * T0)
-(1-q)*T[2,1,0,2] - (q-q^2)*T[1,0] + q^2*T[0]
sage: (T2*T1*T2) * (T2*T1*T0*T1*T2)
-(q-q^2)*T[2,1,0,1,2] + (1-2*q+q^2)*T[2,1,0,2,1,2]
 - (q-q^2)*T[1,0,2,1,2] + q^2*T[0,2,1,2]

Sage Live

T = algebras.ArikiKoike(2, 3).T()
T0, T1, T2 = T.T()
T.product_on_basis(T0.leading_support(), T1.leading_support())
T1 * T2
T2 * T1
T2 * (T2 * T1 * T0)
(T1 * T0 * T1 * T0) * T0
(T0 * T1 * T0 * T1) * (T0 * T1)
T1 * (T0 * T2 * T1 * T0)
(T1 * T2) * (T2 * T1 * T0)
(T2*T1*T2) * (T2*T1*T0*T1*T2)

Python

>>> from sage.all import *
>>> T = algebras.ArikiKoike(Integer(2), Integer(3)).T()
>>> T0, T1, T2 = T.T()
>>> T.product_on_basis(T0.leading_support(), T1.leading_support())
T[0,1]
>>> T1 * T2
T[1,2]
>>> T2 * T1
T[2,1]
>>> T2 * (T2 * T1 * T0)
-(1-q)*T[2,1,0] + q*T[1,0]
>>> (T1 * T0 * T1 * T0) * T0
(-u0*u1)*T[1,0,1] + ((u0+u1))*T[0,1,0,1]
>>> (T0 * T1 * T0 * T1) * (T0 * T1)
(-u0*u1*q)*T[1,0] + (u0*u1-u0*u1*q)*T[1,0,1]
 + ((u0+u1)*q)*T[0,1,0] + ((-u0-u1)+(u0+u1)*q)*T[0,1,0,1]
>>> T1 * (T0 * T2 * T1 * T0)
T[1,0,2,1,0]
>>> (T1 * T2) * (T2 * T1 * T0)
-(1-q)*T[2,1,0,2] - (q-q^2)*T[1,0] + q^2*T[0]
>>> (T2*T1*T2) * (T2*T1*T0*T1*T2)
-(q-q^2)*T[2,1,0,1,2] + (1-2*q+q^2)*T[2,1,0,2,1,2]
 - (q-q^2)*T[1,0,2,1,2] + q^2*T[0,2,1,2]

We check some relations:

Sage

sage: T0 * T1 * T0 * T1 == T1 * T0 * T1 * T0
True
sage: T1 * T2 * T1 == T2 * T1 * T2
True
sage: (T1 * T0) * T0 == T1 * (T0 * T0)
True
sage: (T.L(1) * T.L(2)) * T.L(2) - T.L(1) * (T.L(2) * T.L(2))
0
sage: (T.L(2) * T.L(3)) * T.L(3) - T.L(2) * (T.L(3) * T.L(3))
0

Sage Live

T0 * T1 * T0 * T1 == T1 * T0 * T1 * T0
T1 * T2 * T1 == T2 * T1 * T2
(T1 * T0) * T0 == T1 * (T0 * T0)
(T.L(1) * T.L(2)) * T.L(2) - T.L(1) * (T.L(2) * T.L(2))
(T.L(2) * T.L(3)) * T.L(3) - T.L(2) * (T.L(3) * T.L(3))

Python

>>> from sage.all import *
>>> T0 * T1 * T0 * T1 == T1 * T0 * T1 * T0
True
>>> T1 * T2 * T1 == T2 * T1 * T2
True
>>> (T1 * T0) * T0 == T1 * (T0 * T0)
True
>>> (T.L(Integer(1)) * T.L(Integer(2))) * T.L(Integer(2)) - T.L(Integer(1)) * (T.L(Integer(2)) * T.L(Integer(2)))
0
>>> (T.L(Integer(2)) * T.L(Integer(3))) * T.L(Integer(3)) - T.L(Integer(2)) * (T.L(Integer(3)) * T.L(Integer(3)))
0

a_realization()[source]#

Return a realization of self.

EXAMPLES:

Sage

sage: H = algebras.ArikiKoike(5, 2)
sage: H.a_realization()
Ariki-Koike algebra of rank 5 and order 2
 with q=q and u=(u0, u1, u2, u3, u4) ... in the LT-basis

Sage Live

H = algebras.ArikiKoike(5, 2)
H.a_realization()

Python

>>> from sage.all import *
>>> H = algebras.ArikiKoike(Integer(5), Integer(2))
>>> H.a_realization()
Ariki-Koike algebra of rank 5 and order 2
 with q=q and u=(u0, u1, u2, u3, u4) ... in the LT-basis

cyclotomic_parameters()[source]#

Return the cyclotomic parameters $u$ of self.

EXAMPLES:

Sage

sage: H = algebras.ArikiKoike(5, 3)
sage: H.cyclotomic_parameters()
(u0, u1, u2, u3, u4)

Sage Live

H = algebras.ArikiKoike(5, 3)
H.cyclotomic_parameters()

Python

>>> from sage.all import *
>>> H = algebras.ArikiKoike(Integer(5), Integer(3))
>>> H.cyclotomic_parameters()
(u0, u1, u2, u3, u4)

hecke_parameter()[source]#

Return the Hecke parameter $q$ of self.

EXAMPLES:

Sage

sage: H = algebras.ArikiKoike(5, 3)
sage: H.hecke_parameter()
q

Sage Live

H = algebras.ArikiKoike(5, 3)
H.hecke_parameter()

Python

>>> from sage.all import *
>>> H = algebras.ArikiKoike(Integer(5), Integer(3))
>>> H.hecke_parameter()
q

q()[source]#

Return the Hecke parameter $q$ of self.

EXAMPLES:

Sage

sage: H = algebras.ArikiKoike(5, 3)
sage: H.hecke_parameter()
q

Sage Live

H = algebras.ArikiKoike(5, 3)
H.hecke_parameter()

Python

>>> from sage.all import *
>>> H = algebras.ArikiKoike(Integer(5), Integer(3))
>>> H.hecke_parameter()
q

specht_module(la)[source]#

Return the Specht module of self corresponding to the shape la.

EXAMPLES:

Sage

sage: AK = algebras.ArikiKoike(4, 6)
sage: AK.specht_module([[2], [], [1,1,1], [1]])
Specht module of shape ([2], [], [1, 1, 1], [1]) for
 Ariki-Koike algebra of rank 4 and order 6 with q=q and u=(u0, u1, u2, u3)
 over ... over Integer Ring

Sage Live

AK = algebras.ArikiKoike(4, 6)
AK.specht_module([[2], [], [1,1,1], [1]])

Python

>>> from sage.all import *
>>> AK = algebras.ArikiKoike(Integer(4), Integer(6))
>>> AK.specht_module([[Integer(2)], [], [Integer(1),Integer(1),Integer(1)], [Integer(1)]])
Specht module of shape ([2], [], [1, 1, 1], [1]) for
 Ariki-Koike algebra of rank 4 and order 6 with q=q and u=(u0, u1, u2, u3)
 over ... over Integer Ring

u()[source]#

Return the cyclotomic parameters $u$ of self.

EXAMPLES:

Sage

sage: H = algebras.ArikiKoike(5, 3)
sage: H.cyclotomic_parameters()
(u0, u1, u2, u3, u4)

Sage Live

H = algebras.ArikiKoike(5, 3)
H.cyclotomic_parameters()

Python

>>> from sage.all import *
>>> H = algebras.ArikiKoike(Integer(5), Integer(3))
>>> H.cyclotomic_parameters()
(u0, u1, u2, u3, u4)