Fock Space#

AUTHORS:

Travis Scrimshaw (2013-05-03): Initial version

class sage.algebras.quantum_groups.fock_space.FockSpace(n, multicharge, q, base_ring)#

Bases: Parent, UniqueRepresentation

The (fermionic) Fock space of \(U_q(\widehat{\mathfrak{sl}}_n)\) with multicharge \((\gamma_1, \ldots, \gamma_m)\).

Fix a positive integer \(n > 1\) and fix a sequence \(\gamma = (\gamma_1, \ldots, \gamma_m)\), where \(\gamma_i \in \ZZ / n \ZZ\). (fermionic) Fock space \(\mathcal{F}\) with multicharge \(\gamma\) is a \(U_q(\widehat{\mathfrak{gl}}_n)\)-representation with a basis \(\{ |\lambda \rangle \}\), where \(\lambda\) is a partition tuple of level \(m\). By considering \(\mathcal{F}\) as a \(U_q(\widehat{\mathfrak{sl}}_n)\)-representation, it is not irreducible, but the submodule generated by \(| \emptyset^m \rangle\) is isomorphic to the highest weight module \(V(\mu)\), where the highest weight \(\mu = \sum_i \Lambda_{\gamma_i}\).

Let \(R_i(\lambda)\) and \(A_i(\lambda)\) be the set of removable and addable, respectively, \(i\)-cells of \(\lambda\), where an \(i\)-cell is a cell of residue \(i\) (i.e., content modulo n). The action of \(U_q(\widehat{\mathfrak{sl}}_n)\) is given as follows:

\[\begin{split}\begin{aligned} e_i | \lambda \rangle & = \sum_{c \in R_i(\lambda)} q^{M_i(\lambda, c)} | \lambda + c \rangle, \\ f_i | \lambda \rangle & = \sum_{c \in A_i(\lambda)} q^{N_i(\lambda, c)} | \lambda - c \rangle, \\ q^{h_i} | \lambda \rangle & = q^{N_i(\lambda)} | \lambda \rangle, \\ q^d | \lambda \rangle & = q^{-N^{(0)}(\lambda)} | \lambda \rangle, \end{aligned}\end{split}\]

where

\(M_i(\lambda, c)\) (resp. \(N_i(\lambda, c)\)) is the number of removable (resp. addable) \(i\)-cells of \(\lambda\) below (resp. above) \(c\) minus the number of addable (resp. removable) \(i\)-cells of \(\lambda\) below (resp. above) \(c\),
\(N_i(\lambda)\) is the number of addable \(i\)-cells minus the number of removable \(i\)-cells, and
\(N^{(0)}(\lambda)\) is the total number of \(0\)-cells of \(\lambda\).

Another interpretation of Fock space is as a semi-infinite wedge product (which each factor we can think of as fermions). This allows a description of the \(U_q(\widehat{\mathfrak{gl}}_n)\) action, as well as an explicit description of the bar involution. In particular, the bar involution is the unique semi-linear map satisfying

\(q \mapsto q^{-1}\),
\(\overline{| \emptyset \rangle} = | \emptyset \rangle\), and
\(\overline{f_i | \lambda \rangle} = f_i \overline{| \lambda \rangle}\).

We then define the canonical basis or (lower) global crystal basis as the unique basis of \(\mathcal{F}\) such that

\(\overline{G(\lambda)} = G(\lambda)\),
\(G(\lambda) \equiv | \lambda \rangle \mod q \ZZ[q]\).

It is also known that this basis is upper unitriangular with respect to dominance order and that both the natural basis and the canonical basis of \(\mathcal{F}\) are \(\ZZ\)-graded by \(|\lambda|\). Additionally, the transition matrices \((d_{\lambda, \nu})_{\lambda,\nu \vdash n}\) given by

\[G(\nu) = \sum_{\lambda \vdash |\nu|} d_{\lambda,\nu} |\lambda \rangle\]

described the decomposition matrices of the Hecke algebras when restricting to \(V(\mu)\) [Ariki1996].

To go between the canonical basis and the natural basis, for level 1 Fock space, we follow the LLT algorithm [LLT1996]. Indeed, we first construct a basis \(\{ A(\nu) \}\) that is an approximation to the lower global crystal basis, in the sense that it is bar-invariant, and then use Gaussian elimination to construct the lower global crystal basis. For higher level Fock space, we follow [Fayers2010], where the higher level is considered as a tensor product space of the corresponding level 1 Fock spaces.

There are three bases currently implemented:

The natural basis: F.
The approximation basis that comes from LLT(-type) algorithms: A.
The lower global crystal basis: G.

Todo

Implement the approximation and lower global crystal bases on all partition tuples.
Implement the bar involution.
Implement the full \(U_q(\widehat{\mathfrak{gl}})\)-action.

INPUT:

n – the value \(n\)
multicharge – (default: [0]) the multicharge
q – (optional) the parameter \(q\)
base_ring – (optional) the base ring containing q

EXAMPLES:

We start by constructing the natural basis and doing some computations:

sage: Fock = FockSpace(3)
sage: F = Fock.natural()
sage: u = F.highest_weight_vector()
sage: u.f(0,2,(1,2),0)
|2, 2, 1> + q*|2, 1, 1, 1>
sage: u.f(0,2,(1,2),0,2)
|3, 2, 1> + q*|3, 1, 1, 1> + q*|2, 2, 2> + q^2*|2, 1, 1, 1, 1>
sage: x = u.f(0,2,(1,2),0,2)
sage: [x.h(i) for i in range(3)]
[q*|3, 2, 1> + q^2*|3, 1, 1, 1> + q^2*|2, 2, 2> + q^3*|2, 1, 1, 1, 1>,
 |3, 2, 1> + q*|3, 1, 1, 1> + q*|2, 2, 2> + q^2*|2, 1, 1, 1, 1>,
 |3, 2, 1> + q*|3, 1, 1, 1> + q*|2, 2, 2> + q^2*|2, 1, 1, 1, 1>]
sage: [x.h_inverse(i) for i in range(3)]
[1/q*|3, 2, 1> + |3, 1, 1, 1> + |2, 2, 2> + q*|2, 1, 1, 1, 1>,
 |3, 2, 1> + q*|3, 1, 1, 1> + q*|2, 2, 2> + q^2*|2, 1, 1, 1, 1>,
 |3, 2, 1> + q*|3, 1, 1, 1> + q*|2, 2, 2> + q^2*|2, 1, 1, 1, 1>]
sage: x.d()
1/q^2*|3, 2, 1> + 1/q*|3, 1, 1, 1> + 1/q*|2, 2, 2> + |2, 1, 1, 1, 1>

Next, we construct the approximation and lower global crystal bases and convert to the natural basis:

sage: A = Fock.A()
sage: G = Fock.G()
sage: F(A[4,2,2,1])
|4, 2, 2, 1> + q*|4, 2, 1, 1, 1>
sage: F(G[4,2,2,1])
|4, 2, 2, 1> + q*|4, 2, 1, 1, 1>
sage: F(A[7,3,2,1,1])
|7, 3, 2, 1, 1> + q*|7, 2, 2, 2, 1> + q^2*|7, 2, 2, 1, 1, 1>
 + q*|6, 3, 3, 1, 1> + q^2*|6, 2, 2, 2, 2> + q^3*|6, 2, 2, 1, 1, 1, 1>
 + q*|5, 5, 2, 1, 1> + q^2*|5, 4, 3, 1, 1> + (q^2+1)*|4, 4, 3, 2, 1>
 + (q^3+q)*|4, 4, 3, 1, 1, 1> + (q^3+q)*|4, 4, 2, 2, 2>
 + (q^4+q^2)*|4, 4, 2, 1, 1, 1, 1> + q*|4, 3, 3, 3, 1>
 + q^2*|4, 3, 2, 1, 1, 1, 1, 1> + q^2*|4, 2, 2, 2, 2, 2>
 + q^3*|4, 2, 2, 2, 1, 1, 1, 1> + q^2*|3, 3, 3, 3, 2>
 + q^3*|3, 3, 3, 1, 1, 1, 1, 1> + q^3*|3, 2, 2, 2, 2, 2, 1>
 + q^4*|3, 2, 2, 2, 2, 1, 1, 1>
sage: F(G[7,3,2,1,1])
|7, 3, 2, 1, 1> + q*|7, 2, 2, 2, 1> + q^2*|7, 2, 2, 1, 1, 1>
 + q*|6, 3, 3, 1, 1> + q^2*|6, 2, 2, 2, 2>
 + q^3*|6, 2, 2, 1, 1, 1, 1> + q*|5, 5, 2, 1, 1>
 + q^2*|5, 4, 3, 1, 1> + q^2*|4, 4, 3, 2, 1>
 + q^3*|4, 4, 3, 1, 1, 1> + q^3*|4, 4, 2, 2, 2>
 + q^4*|4, 4, 2, 1, 1, 1, 1>
sage: A(F(G[7,3,2,1,1]))
A[7, 3, 2, 1, 1] - A[4, 4, 3, 2, 1]
sage: G(F(A[7,3,2,1,1]))
G[7, 3, 2, 1, 1] + G[4, 4, 3, 2, 1]
sage: A(F(G[8,4,3,2,2,1]))
A[8, 4, 3, 2, 2, 1] - A[6, 4, 4, 2, 2, 1, 1] - A[5, 5, 4, 3, 2, 1]
 + ((-q^2-1)/q)*A[5, 4, 4, 3, 2, 1, 1]
sage: G(F(A[8,4,3,2,2,1]))
G[8, 4, 3, 2, 2, 1] + G[6, 4, 4, 2, 2, 1, 1] + G[5, 5, 4, 3, 2, 1]
 + ((q^2+1)/q)*G[5, 4, 4, 3, 2, 1, 1]

We can also construct higher level Fock spaces and perform similar computations:

sage: Fock = FockSpace(3, [1,0])
sage: F = Fock.natural()
sage: A = Fock.A()
sage: G = Fock.G()
sage: F(G[[2,1],[4,1,1]])
|[2, 1], [4, 1, 1]> + q*|[2, 1], [3, 2, 1]>
 + q^2*|[2, 1], [3, 1, 1, 1]> + q^2*|[2], [4, 2, 1]>
 + q^3*|[2], [4, 1, 1, 1]> + q^4*|[2], [3, 2, 1, 1]>
 + q*|[1, 1, 1], [4, 1, 1]> + q^2*|[1, 1, 1], [3, 2, 1]>
 + q^3*|[1, 1, 1], [3, 1, 1, 1]> + q^2*|[1, 1], [3, 2, 2]>
 + q^3*|[1, 1], [3, 1, 1, 1, 1]> + q^3*|[1], [4, 2, 2]>
 + q^4*|[1], [4, 1, 1, 1, 1]> + q^4*|[1], [3, 2, 2, 1]>
 + q^5*|[1], [3, 2, 1, 1, 1]>
sage: A(F(G[[2,1],[4,1,1]]))
A([2, 1], [4, 1, 1]) - A([2], [4, 2, 1])
sage: G(F(A[[2,1],[4,1,1]]))
G([2, 1], [4, 1, 1]) + G([2], [4, 2, 1])

For level \(0\), the truncated Fock space of [GW1999] is implemented. This can be used to improve the speed of the computation of the lower global crystal basis, provided the truncation is not too small:

sage: FS = FockSpace(2)
sage: F = FS.natural()
sage: G = FS.G()
sage: FS3 = FockSpace(2, truncated=3)
sage: F3 = FS3.natural()
sage: G3 = FS3.G()
sage: F(G[6,2,1])
|6, 2, 1> + q*|5, 3, 1> + q^2*|5, 2, 2> + q^3*|5, 2, 1, 1>
 + q*|4, 2, 1, 1, 1> + q^2*|3, 3, 1, 1, 1> + q^3*|3, 2, 2, 1, 1>
 + q^4*|3, 2, 1, 1, 1, 1>
sage: F3(G3[6,2,1])
|6, 2, 1> + q*|5, 3, 1> + q^2*|5, 2, 2>
sage: FS5 = FockSpace(2, truncated=5)
sage: F5 = FS5.natural()
sage: G5 = FS5.G()
sage: F5(G5[6,2,1])
|6, 2, 1> + q*|5, 3, 1> + q^2*|5, 2, 2> + q^3*|5, 2, 1, 1>
 + q*|4, 2, 1, 1, 1> + q^2*|3, 3, 1, 1, 1> + q^3*|3, 2, 2, 1, 1>

REFERENCES:

class A(F)#

Bases: CombinatorialFreeModule, BindableClass

The \(A\) basis of the Fock space which is the approximation of the lower global crystal basis.

The approximation basis \(A\) is a basis that is constructed from the highest weight element by applying divided difference operators using the ladder construction of [LLT1996] and [GW1999]. Thus, this basis is bar invariant and upper unitriangular (using dominance order on partitions) when expressed in the natural basis. This basis is then converted to the lower global crystal basis by using Gaussian elimination.

EXAMPLES:

We construct Example 6.5 and 6.7 in [LLT1996]:

sage: FS = FockSpace(2)
sage: F = FS.natural()
sage: G = FS.G()
sage: A = FS.A()
sage: F(A[5])
|5> + |3, 2> + 2*q*|3, 1, 1> + q^2*|2, 2, 1> + q^2*|1, 1, 1, 1, 1>
sage: F(A[4,1])
|4, 1> + q*|2, 1, 1, 1>
sage: F(A[3,2])
|3, 2> + q*|3, 1, 1> + q^2*|2, 2, 1>
sage: F(G[5])
|5> + q*|3, 1, 1> + q^2*|1, 1, 1, 1, 1>

We construct the examples in Section 5.1 of [Fayers2010]:

sage: FS = FockSpace(2, [0, 0])
sage: F = FS.natural()
sage: A = FS.A()
sage: F(A[[2,1],[1]])
|[2, 1], [1]> + q*|[2], [2]> + q^2*|[2], [1, 1]> + q^2*|[1, 1], [2]>
 + q^3*|[1, 1], [1, 1]> + q^4*|[1], [2, 1]>
sage: F(A[[4],[]])
|[4], []> + q*|[3, 1], []> + q*|[2, 1, 1], []>
 + (q^2+1)*|[2, 1], [1]> + 2*q*|[2], [2]> + 2*q^2*|[2], [1, 1]>
 + q^2*|[1, 1, 1, 1], []> + 2*q^2*|[1, 1], [2]>
 + 2*q^3*|[1, 1], [1, 1]> + (q^4+q^2)*|[1], [2, 1]>
 + q^2*|[], [4]> + q^3*|[], [3, 1]> + q^3*|[], [2, 1, 1]>
 + q^4*|[], [1, 1, 1, 1]>

options = Current options for FockSpace - display: ket#

class F(F)#

Bases: CombinatorialFreeModule, BindableClass

The natural basis of the Fock space.

This is the basis indexed by partitions. This has an action of the quantum group \(U_q(\widehat{\mathfrak{sl}}_n)\) described in FockSpace.

EXAMPLES:

We construct the natural basis and perform some computations:

sage: F = FockSpace(4).natural()
sage: q = F.q()
sage: u = F.highest_weight_vector()
sage: u
|>
sage: u.f(0,1,2)
|3>
sage: u.f(0,1,3)
|2, 1>
sage: u.f(0,1,2,0)
0
sage: u.f(0,1,3,2)
|3, 1> + q*|2, 1, 1>
sage: u.f(0,1,2,3)
|4> + q*|3, 1>
sage: u.f(0,1,3,2,2,0)
((q^2+1)/q)*|3, 2, 1>
sage: x = (q^4 * u + u.f(0,1,3,(2,2)))
sage: x
|3, 1, 1> + q^4*|>
sage: x.f(0,1,3)
|4, 3, 1> + q*|4, 2, 1, 1> + q*|3, 3, 2>
 + q^2*|3, 2, 2, 1> + q^4*|2, 1>
sage: x.h_inverse(2)
q^2*|3, 1, 1> + q^4*|>
sage: x.h_inverse(0)
1/q*|3, 1, 1> + q^3*|>
sage: x.d()
1/q*|3, 1, 1> + q^4*|>
sage: x.e(2)
|3, 1> + q*|2, 1, 1>

class Element#

Bases: IndexedFreeModuleElement

An element in the Fock space.

d()#

Apply the action of \(d\) on self.

EXAMPLES:

sage: F = FockSpace(2)
sage: F.highest_weight_vector().d()
|>
sage: F[2,1,1].d()
1/q^2*|2, 1, 1>
sage: F[5,3,3,1,1,1].d()
1/q^7*|5, 3, 3, 1, 1, 1>

sage: F = FockSpace(4, [2,0,1])
sage: F.highest_weight_vector().d()
|[], [], []>
sage: F[[2,1],[1],[2]].d()
1/q*|[2, 1], [1], [2]>
sage: F[[4,2,2,1],[1],[5,2]].d()
1/q^5*|[4, 2, 2, 1], [1], [5, 2]>

e(*data)#

Apply the action of the divided power operator \(e_i^{(p)} = e_i^{p} / [p]_q\) on self.

INPUT:

*data – a list of indices or pairs \((i, p)\)

EXAMPLES:

sage: F = FockSpace(2)
sage: F[2,1,1].e(1)
1/q*|1, 1, 1>
sage: F[2,1,1].e(0)
|2, 1>
sage: F[2,1,1].e(0).e(1)
|2> + q*|1, 1>
sage: F[2,1,1].e(0).e(1).e(1)
((q^2+1)/q)*|1>
sage: F[2,1,1].e(0).e((1, 2))
|1>
sage: F[2,1,1].e(0, 1, 1, 1)
0
sage: F[2,1,1].e(0, (1, 3))
0
sage: F[2,1,1].e(0, (1,2), 0)
|>
sage: F[2,1,1].e(1, 0, 1, 0)
1/q*|>

sage: F = FockSpace(4, [2, 0, 1])
sage: F[[2,1],[1],[2]]
|[2, 1], [1], [2]>
sage: F[[2,1],[1],[2]].e(2)
|[2, 1], [1], [1]>
sage: F[[2,1],[1],[2]].e(1)
1/q*|[2], [1], [2]>
sage: F[[2,1],[1],[2]].e(0)
1/q*|[2, 1], [], [2]>
sage: F[[2,1],[1],[2]].e(3)
1/q^2*|[1, 1], [1], [2]>
sage: F[[2,1],[1],[2]].e(3, 2, 1)
1/q^2*|[1, 1], [1], []> + 1/q^2*|[1], [1], [1]>
sage: F[[2,1],[1],[2]].e(3, 2, 1, 0, 1, 2)
2/q^3*|[], [], []>

f(*data)#

Apply the action of the divided power operator \(f_i^{(p)} = f_i^{p} / [p]_q\) on self.

INPUT:

*data – a list of indices or pairs \((i, p)\)

EXAMPLES:

sage: F = FockSpace(2)
sage: mg = F.highest_weight_vector()
sage: mg.f(0)
|1>
sage: mg.f(0).f(1)
|2> + q*|1, 1>
sage: mg.f(0).f(0)
0
sage: mg.f((0, 2))
0
sage: mg.f(0, 1, 1)
((q^2+1)/q)*|2, 1>
sage: mg.f(0, (1, 2))
|2, 1>
sage: mg.f(0, 1, 0)
|3> + q*|1, 1, 1>

sage: F = FockSpace(4, [2, 0, 1])
sage: mg = F.highest_weight_vector()
sage: mg.f(0)
|[], [1], []>
sage: mg.f(2)
|[1], [], []>
sage: mg.f(1)
|[], [], [1]>
sage: mg.f(1, 0)
|[], [1], [1]> + q*|[], [], [1, 1]>
sage: mg.f(0, 1)
|[], [2], []> + q*|[], [1], [1]>
sage: mg.f(0, 1, 3)
|[], [2, 1], []> + q*|[], [1, 1], [1]>
sage: mg.f(3)
0

h(*data)#

Apply the action of \(h_i\) on self.

EXAMPLES:

sage: F = FockSpace(2)
sage: F[2,1,1].h(0)
q*|2, 1, 1>
sage: F[2,1,1].h(1)
|2, 1, 1>
sage: F[2,1,1].h(0, 0)
q^2*|2, 1, 1>

sage: F = FockSpace(4, [2,0,1])
sage: elt = F[[2,1],[1],[2]]
sage: elt.h(0)
q^2*|[2, 1], [1], [2]>
sage: elt.h(1)
|[2, 1], [1], [2]>
sage: elt.h(2)
|[2, 1], [1], [2]>
sage: elt.h(3)
q*|[2, 1], [1], [2]>

h_inverse(*data)#

Apply the action of \(h_i^{-1}\) on self.

EXAMPLES:

sage: F = FockSpace(2)
sage: F[2,1,1].h_inverse(0)
1/q*|2, 1, 1>
sage: F[2,1,1].h_inverse(1)
|2, 1, 1>
sage: F[2,1,1].h_inverse(0, 0)
1/q^2*|2, 1, 1>

sage: F = FockSpace(4, [2,0,1])
sage: elt = F[[2,1],[1],[2]]
sage: elt.h_inverse(0)
1/q^2*|[2, 1], [1], [2]>
sage: elt.h_inverse(1)
|[2, 1], [1], [2]>
sage: elt.h_inverse(2)
|[2, 1], [1], [2]>
sage: elt.h_inverse(3)
1/q*|[2, 1], [1], [2]>

options = Current options for FockSpace - display: ket#

class G(F)#

Bases: CombinatorialFreeModule, BindableClass

The lower global crystal basis living inside of Fock space.

EXAMPLES:

We construct some of the tables/entries given in Section 10 of [LLT1996]. For \(\widehat{\mathfrak{sl}}_2\):

sage: FS = FockSpace(2)
sage: F = FS.natural()
sage: G = FS.G()
sage: F(G[2])
|2> + q*|1, 1>
sage: F(G[3])
|3> + q*|1, 1, 1>
sage: F(G[2,1])
|2, 1>
sage: F(G[4])
|4> + q*|3, 1> + q*|2, 1, 1> + q^2*|1, 1, 1, 1>
sage: F(G[3,1])
|3, 1> + q*|2, 2> + q^2*|2, 1, 1>
sage: F(G[5])
|5> + q*|3, 1, 1> + q^2*|1, 1, 1, 1, 1>
sage: F(G[4,2])
|4, 2> + q*|4, 1, 1> + q*|3, 3> + q^2*|3, 1, 1, 1>
 + q^2*|2, 2, 2> + q^3*|2, 2, 1, 1>
sage: F(G[4,2,1])
|4, 2, 1> + q*|3, 3, 1> + q^2*|3, 2, 2> + q^3*|3, 2, 1, 1>
sage: F(G[6,2])
|6, 2> + q*|6, 1, 1> + q*|5, 3> + q^2*|5, 1, 1, 1> + q*|4, 3, 1>
 + q^2*|4, 2, 2> + (q^3+q)*|4, 2, 1, 1> + q^2*|4, 1, 1, 1, 1>
 + q^2*|3, 3, 1, 1> + q^3*|3, 2, 2, 1> + q^3*|3, 1, 1, 1, 1, 1>
 + q^3*|2, 2, 2, 1, 1> + q^4*|2, 2, 1, 1, 1, 1>
sage: F(G[5,3,1])
|5, 3, 1> + q*|5, 2, 2> + q^2*|5, 2, 1, 1> + q*|4, 4, 1>
 + q^2*|4, 2, 1, 1, 1> + q^2*|3, 3, 3> + q^3*|3, 3, 1, 1, 1>
 + q^3*|3, 2, 2, 2> + q^4*|3, 2, 2, 1, 1>
sage: F(G[4,3,2,1])
|4, 3, 2, 1>
sage: F(G[7,2,1])
|7, 2, 1> + q*|5, 2, 1, 1, 1> + q^2*|3, 2, 1, 1, 1, 1, 1>
sage: F(G[10,1])
|10, 1> + q*|8, 1, 1, 1> + q^2*|6, 1, 1, 1, 1, 1>
 + q^3*|4, 1, 1, 1, 1, 1, 1, 1>
 + q^4*|2, 1, 1, 1, 1, 1, 1, 1, 1, 1>
sage: F(G[6,3,2])
|6, 3, 2> + q*|6, 3, 1, 1> + q^2*|6, 2, 2, 1> + q^3*|5, 3, 2, 1>
 + q*|4, 3, 2, 1, 1> + q^2*|4, 3, 1, 1, 1, 1>
 + q^3*|4, 2, 2, 1, 1, 1> + q^4*|3, 3, 2, 1, 1, 1>
sage: F(G[5,3,2,1])
|5, 3, 2, 1> + q*|4, 4, 2, 1> + q^2*|4, 3, 3, 1>
 + q^3*|4, 3, 2, 2> + q^4*|4, 3, 2, 1, 1>

For \(\widehat{\mathfrak{sl}}_3\):

sage: FS = FockSpace(3)
sage: F = FS.natural()
sage: G = FS.G()
sage: F(G[2])
|2>
sage: F(G[1,1])
|1, 1>
sage: F(G[3])
|3> + q*|2, 1>
sage: F(G[2,1])
|2, 1> + q*|1, 1, 1>
sage: F(G[4])
|4> + q*|2, 2>
sage: F(G[3,1])
|3, 1>
sage: F(G[2,2])
|2, 2> + q*|1, 1, 1, 1>
sage: F(G[2,1,1])
|2, 1, 1>
sage: F(G[5])
|5> + q*|2, 2, 1>
sage: F(G[2,2,1])
|2, 2, 1> + q*|2, 1, 1, 1>
sage: F(G[4,1,1])
|4, 1, 1> + q*|3, 2, 1> + q^2*|3, 1, 1, 1>
sage: F(G[5,2])
|5, 2> + q*|4, 3> + q^2*|4, 2, 1>
sage: F(G[8])
|8> + q*|5, 2, 1> + q*|3, 3, 1, 1> + q^2*|2, 2, 2, 2>
sage: F(G[7,2])
|7, 2> + q*|4, 2, 2, 1>
sage: F(G[6,2,2])
|6, 2, 2> + q*|6, 1, 1, 1, 1> + q*|4, 4, 2> + q^2*|3, 3, 2, 1, 1>

For \(\widehat{\mathfrak{sl}}_4\):

sage: FS = FockSpace(4)
sage: F = FS.natural()
sage: G = FS.G()
sage: F(G[4])
|4> + q*|3, 1>
sage: F(G[3,1])
|3, 1> + q*|2, 1, 1>
sage: F(G[2,2])
|2, 2>
sage: F(G[2,1,1])
|2, 1, 1> + q*|1, 1, 1, 1>
sage: F(G[3,2])
|3, 2> + q*|2, 2, 1>
sage: F(G[2,2,2])
|2, 2, 2> + q*|1, 1, 1, 1, 1, 1>
sage: F(G[6,1])
|6, 1> + q*|4, 3>
sage: F(G[3,2,2,1])
|3, 2, 2, 1> + q*|3, 1, 1, 1, 1, 1> + q*|2, 2, 2, 2>
 + q^2*|2, 1, 1, 1, 1, 1, 1>
sage: F(G[7,2])
|7, 2> + q*|6, 2, 1> + q*|5, 4> + q^2*|5, 3, 1>
sage: F(G[5,2,2,1])
|5, 2, 2, 1> + q*|5, 1, 1, 1, 1, 1> + q*|4, 2, 2, 1, 1>
 + q^2*|4, 2, 1, 1, 1, 1>

We construct the examples in Section 5.1 of [Fayers2010]:

sage: FS = FockSpace(2, [0, 0])
sage: F = FS.natural()
sage: G = FS.G()
sage: F(G[[2,1],[1]])
|[2, 1], [1]> + q*|[2], [2]> + q^2*|[2], [1, 1]>
 + q^2*|[1, 1], [2]> + q^3*|[1, 1], [1, 1]> + q^4*|[1], [2, 1]>
sage: F(G[[4],[]])
|[4], []> + q*|[3, 1], []> + q*|[2, 1, 1], []> + q^2*|[2, 1], [1]>
 + q*|[2], [2]> + q^2*|[2], [1, 1]> + q^2*|[1, 1, 1, 1], []>
 + q^2*|[1, 1], [2]> + q^3*|[1, 1], [1, 1]> + q^2*|[1], [2, 1]>
 + q^2*|[], [4]> + q^3*|[], [3, 1]> + q^3*|[], [2, 1, 1]>
 + q^4*|[], [1, 1, 1, 1]>

options = Current options for FockSpace - display: ket#

a_realization()#

Return a realization of self.

EXAMPLES:

sage: FS = FockSpace(2)
sage: FS.a_realization()
Fock space of rank 2 of multicharge (0,) over
 Fraction Field of Univariate Polynomial Ring in q over Integer Ring
 in the natural basis

approximation#: alias of A

canonical#: alias of G

highest_weight_vector()#

Return the module generator of self in the natural basis.

EXAMPLES:

sage: FS = FockSpace(2)
sage: FS.highest_weight_vector()
|>
sage: FS = FockSpace(4, [2, 0, 1])
sage: FS.highest_weight_vector()
|[], [], []>

inject_shorthands(verbose=True)#

Import standard shorthands into the global namespace.

INPUT:

verbose – boolean (default True) if True, prints the defined shorthands

EXAMPLES:

sage: FS = FockSpace(4)
sage: FS.inject_shorthands()
Injecting A as shorthand for Fock space of rank 4
 of multicharge (0,) over Fraction Field
 of Univariate Polynomial Ring in q over Integer Ring
 in the approximation basis
Injecting F as shorthand for Fock space of rank 4
 of multicharge (0,) over Fraction Field
 of Univariate Polynomial Ring in q over Integer Ring
 in the natural basis
Injecting G as shorthand for Fock space of rank 4
 of multicharge (0,) over Fraction Field
 of Univariate Polynomial Ring in q over Integer Ring
 in the lower global crystal basis

lower_global_crystal#: alias of G

multicharge()#

Return the multicharge of self.

EXAMPLES:

sage: F = FockSpace(2)
sage: F.multicharge()
(0,)

sage: F = FockSpace(4, [2, 0, 1])
sage: F.multicharge()
(2, 0, 1)

natural#: alias of F

options = Current options for FockSpace - display: ket#

q()#

Return the parameter \(q\) of self.

EXAMPLES:

sage: F = FockSpace(2)
sage: F.q()
q

sage: F = FockSpace(2, q=-1)
sage: F.q()
-1

class sage.algebras.quantum_groups.fock_space.FockSpaceBases(base)#

Bases: Category_realization_of_parent

The category of bases of a (truncated) Fock space.

class ParentMethods#

Bases: object

highest_weight_vector()#

Return the highest weight vector of self.

EXAMPLES:

sage: FS = FockSpace(2)
sage: F = FS.natural()
sage: F.highest_weight_vector()
|>
sage: A = FS.A()
sage: A.highest_weight_vector()
A[]
sage: G = FS.G()
sage: G.highest_weight_vector()
G[]

multicharge()#

Return the multicharge of self.

EXAMPLES:

sage: FS = FockSpace(4)
sage: A = FS.A()
sage: A.multicharge()
(0,)

sage: FS = FockSpace(4, [1,0,2])
sage: G = FS.G()
sage: G.multicharge()
(1, 0, 2)

q()#

Return the parameter \(q\) of self.

EXAMPLES:

sage: FS = FockSpace(2)
sage: A = FS.A()
sage: A.q()
q

sage: FS = FockSpace(2, q=-1)
sage: G = FS.G()
sage: G.q()
-1

some_elements()#

Return some elements of self.

EXAMPLES:

sage: F = FockSpace(3).natural()
sage: F.some_elements()[::13]
[3*|2> + 2*|1> + 2*|>,
 |5>,
 |3, 1, 1, 1>,
 |3, 2, 2>,
 |5, 1, 1, 1>,
 |2, 2, 1, 1, 1, 1>,
 |5, 2, 1, 1>,
 |3, 2, 1, 1, 1, 1>]

sage: F = FockSpace(3, [0,1]).natural()
sage: F.some_elements()[::13]
[2*|[1], []> + 4*|[], [1]> + |[], []>,
 |[1, 1], [1]>,
 |[1, 1, 1], [1]>,
 |[5], []>,
 |[3], [1, 1]>,
 |[1], [2, 2]>,
 |[4, 1, 1], []>,
 |[2, 1, 1, 1], [1]>]

super_categories()#

The super categories of self.

EXAMPLES:

sage: from sage.algebras.quantum_groups.fock_space import FockSpaceBases
sage: F = FockSpace(2)
sage: bases = FockSpaceBases(F)
sage: bases.super_categories()
[Category of vector spaces with basis over Fraction Field
  of Univariate Polynomial Ring in q over Integer Ring,
 Category of realizations of Fock space of rank 2 of multicharge (0,)
  over Fraction Field of Univariate Polynomial Ring in q over Integer Ring]

sage.algebras.quantum_groups.fock_space.FockSpaceOptions(*get_value, **set_value)#

Sets and displays the global options for elements of the Fock space classes. If no parameters are set, then the function returns a copy of the options dictionary.

The options to Fock space can be accessed as the method FockSpaceOptions of FockSpace and related parent classes.

OPTIONS:

display – (default: ket) Specifies how terms of the natural basis of Fock space should be printed
- ket – displayed as a ket in bra-ket notation
- list – displayed as a list

EXAMPLES:

sage: FS = FockSpace(4)
sage: F = FS.natural()
sage: x = F.an_element()
sage: y = x.f(3,2,2,0,1)
sage: y
((3*q^2+3)/q)*|3, 3, 1> + (3*q^2+3)*|3, 2, 1, 1>
sage: Partitions.options.display = 'diagram'
sage: y
((3*q^2+3)/q)*|3, 3, 1> + (3*q^2+3)*|3, 2, 1, 1>
sage: ascii_art(y)
((3*q^2+3)/q)*|***\  + (3*q^2+3)*|***\
              |*** >             |**  \
              |*  /              |*   /
                                 |*  /
sage: FockSpace.options.display = 'list'
sage: ascii_art(y)
((3*q^2+3)/q)*F    + (3*q^2+3)*F
               ***              ***
               ***              **
               *                *
                                *
sage: Partitions.options.display = 'compact_high'
sage: y
((3*q^2+3)/q)*F3^2,1 + (3*q^2+3)*F3,2,1^2

sage: Partitions.options._reset()
sage: FockSpace.options._reset()

See GlobalOptions for more features of these options.

class sage.algebras.quantum_groups.fock_space.FockSpaceTruncated(n, k, q, base_ring)#

Bases: FockSpace

This is the Fock space given by partitions of length no more than \(k\).

This can be formed as the quotient \(\mathcal{F} / \mathcal{F}_k\), where \(\mathcal{F}_k\) is the submodule spanned by all diagrams of length (strictly) more than \(k\).

We have three bases:

The natural basis indexed by truncated \(n\)-regular partitions: F.
The approximation basis that comes from LLT(-type) algorithms: A.
The lower global crystal basis: G.