Quantum Group Representations¶
AUTHORS:
- Travis Scrimshaw (2018): initial version
-
sage.algebras.quantum_groups.representations.
AdjointRepresentation
¶ An (generalized) adjoint representation of a quantum group.
We define an (generalized) adjoint representation \(V\) of a quantum group \(U_q\) to be a cyclic \(U_q\)-module with a weight space decomposition \(V = \bigoplus_{\mu} V_{\mu}\) such that \(\dim V_{\mu} \leq 1\) unless \(\mu = 0\). Moreover, we require that there exists a basis \(\{y_j | j \in J\}\) for \(V_0\) such that \(e_i y_j = 0\) for all \(j \neq i \in I\).
For a base ring \(R\), we construct an adjoint representation from its (combinatorial) crystal \(B\) by \(V = R \{v_b | b \in B\}\) with
\[\begin{split}\begin{aligned} e_i v_b & = \begin{cases} v_{e_i b} / [\varphi_i(e_i b)]_{q_i}, & \text{if } \operatorname{wt}(b) \neq 0, \\ v_{e_i b} + \sum_{j \neq i} [-A_{ij}]_{q_i} / [2]_{q_i} v_{y_j} & \text{otherwise} \end{cases} \\ f_i v_b & = \begin{cases} v_{f_i b} / [\varepsilon_i(f_i b)]_{q_i}, & \text{if } \operatorname{wt}(b) \neq 0, \\ v_{f_i b} + \sum_{j \neq i} [-A_{ij}]_{q_i} / [2]_{q_i} v_{y_j} & \text{otherwise} \end{cases} \\ K_i v_b & = q^{\langle h_i, \operatorname{wt}(b) \rangle} v_b, \end{aligned}\end{split}\]where \((A_{ij})_{i,j \in I}\) is the Cartan matrix, and we consider \(v_0 := 0\).
INPUT:
C
– the crystal corresponding to the representationR
– the base ringq
– (default: the generator ofR
) the parameter \(q\) of the quantum group
Warning
This assumes that \(q\) is generic.
EXAMPLES:
sage: from sage.algebras.quantum_groups.representations import AdjointRepresentation sage: R = ZZ['q'].fraction_field() sage: C = crystals.Tableaux(['D',4], shape=[1,1]) sage: V = AdjointRepresentation(R, C) sage: V V((1, 1, 0, 0)) sage: v = V.an_element(); v 2*B[[[1], [2]]] + 2*B[[[1], [3]]] + 3*B[[[2], [3]]] sage: v.e(2) 2*B[[[1], [2]]] sage: v.f(2) 2*B[[[1], [3]]] sage: v.f(4) 2*B[[[1], [-4]]] + 3*B[[[2], [-4]]] sage: v.K(3) 2*B[[[1], [2]]] + 2*q*B[[[1], [3]]] + 3*q*B[[[2], [3]]] sage: v.K(2,-2) 2/q^2*B[[[1], [2]]] + 2*q^2*B[[[1], [3]]] + 3*B[[[2], [3]]] sage: La = RootSystem(['F',4,1]).weight_space().fundamental_weights() sage: K = crystals.ProjectedLevelZeroLSPaths(La[4]) sage: A = AdjointRepresentation(R, K) sage: A V(-Lambda[0] + Lambda[4]) sage: v = A.an_element(); v 3*B[(-Lambda[0] + Lambda[3] - Lambda[4],)] + 2*B[(Lambda[0] - Lambda[1] + Lambda[4],)] + 2*B[(-Lambda[0] + Lambda[4],)] sage: v.e(0) 2*B[(Lambda[0] - Lambda[1] + Lambda[4],)] + 3*B[(Lambda[0] - Lambda[1] + Lambda[3] - Lambda[4],)] sage: v.f(0) 2*B[(-Lambda[0] + Lambda[4],)]
REFERENCES:
-
sage.algebras.quantum_groups.representations.
CyclicRepresentation
¶ A cyclic quantum group representation that is indexed by either a highest weight crystal or Kirillov-Reshetikhin crystal.
The crystal
C
must either allowC.module_generator()
, otherwise it is assumed to be generated byC.module_generators[0]
.This is meant as an abstract base class for
AdjointRepresentation
andMinusculeRepresentation
.
-
sage.algebras.quantum_groups.representations.
MinusculeRepresentation
¶ A minuscule representation of a quantum group.
A quantum group representation \(V\) is minuscule if it is cyclic, there is a weight space decomposition \(V = \bigoplus_{\mu} V_{\mu}\) with \(\dim V_{\mu} \leq 1\), and \(e_i^2 V = 0\) and \(f_i^2 V = 0\).
For a base ring \(R\), we construct a minuscule representation from its (combinatorial) crystal \(B\) by \(V = R \{v_b | b \in B\}\) with \(e_i v_b = v_{e_i b}\), \(f_i v_b = v_{f_i b}\), and \(K_i v_b = q^{\langle h_i, \operatorname{wt}(b) \rangle} v_b\), where we consider \(v_0 := 0\).
INPUT:
C
– the crystal corresponding to the representationR
– the base ringq
– (default: the generator ofR
) the parameter \(q\) of the quantum group
Warning
This assumes that \(q\) is generic.
EXAMPLES:
sage: from sage.algebras.quantum_groups.representations import MinusculeRepresentation sage: R = ZZ['q'].fraction_field() sage: C = crystals.Tableaux(['B',3], shape=[1/2,1/2,1/2]) sage: V = MinusculeRepresentation(R, C) sage: V V((1/2, 1/2, 1/2)) sage: v = V.an_element(); v 2*B[[+++, []]] + 2*B[[++-, []]] + 3*B[[+-+, []]] sage: v.e(3) 2*B[[+++, []]] sage: v.f(1) 3*B[[-++, []]] sage: v.f(3) 2*B[[++-, []]] + 3*B[[+--, []]] sage: v.K(2) 2*B[[+++, []]] + 2*q^2*B[[++-, []]] + 3/q^2*B[[+-+, []]] sage: v.K(3, -2) 2/q^2*B[[+++, []]] + 2*q^2*B[[++-, []]] + 3/q^2*B[[+-+, []]] sage: K = crystals.KirillovReshetikhin(['D',4,2], 3,1) sage: A = MinusculeRepresentation(R, K) sage: A V(-Lambda[0] + Lambda[3]) sage: v = A.an_element(); v 2*B[[+++, []]] + 2*B[[++-, []]] + 3*B[[+-+, []]] sage: v.f(0) 0 sage: v.e(0) 2*B[[-++, []]] + 2*B[[-+-, []]] + 3*B[[--+, []]]
REFERENCES:
-
sage.algebras.quantum_groups.representations.
QuantumGroupRepresentation
¶ A representation of a quantum group whose basis is indexed by the corresponding (combinatorial) crystal.
INPUT:
C
– the crystal corresponding to the representationR
– the base ringq
– (default: the generator ofR
) the parameter \(q\) of the quantum group