Affinization Crystals

sage.combinat.crystals.affinization.AffinizationOfCrystal

An affinization of a crystal.

Let \(\mathfrak{g}\) be a Kac-Moody algebra of affine type. The affinization of a finite \(U_q^{\prime}(\mathfrak{g})\)-crystal \(B\) is the (infinite) \(U_q(\mathfrak{g})\)-crystal with underlying set:

\[B^{\mathrm{aff}} = \{ b(m) \mid b \in B, m \in \ZZ \}\]

and crystal structure determined by:

\[\begin{split}\begin{aligned} e_i(b(m)) & = \begin{cases} (e_0 b)(m+1) & i = 0, \\ (e_i b)(m) & i \neq 0, \end{cases} \\ f_i(b(m)) &= \begin{cases} (f_0 b)(m-1) & i = 0, \\ (f_i b)(m) & i \neq 0, \end{cases} \\ \mathrm{wt}(b(m)) &= \mathrm{wt}(b) + m \delta. \end{aligned}\end{split}\]

EXAMPLES:

We first construct a Kirillov-Reshetikhin crystal and then take it’s corresponding affinization:

sage: K = crystals.KirillovReshetikhin(['A',2,1], 2, 2)
sage: A = K.affinization()

Next we construct an affinization crystal from a tensor product of KR crystals:

sage: KT = crystals.TensorProductOfKirillovReshetikhinTableaux(['C',2,1], [[1,2],[2,1]])
sage: A = crystals.AffinizationOf(KT)

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