Canonical basis

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In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context:

Representation theory[edit]

In representation theory there are several basis that are called "canonical", e.g. Lusztig's canonical basis and closely related Kashiwara's crystal basis in quantum groups and their representations. There is a general concept underlying these basis:

Consider the ring of integral Laurent polynomials \mathcal{Z}:=\mathbb{Z}[v,v^{-1}] with its two subrings \mathcal{Z}^{\pm}:=\mathbb{Z}[v^{\pm 1}] and the automorphism \overline{\cdot} that is defined by \overline{v}:=v^{-1}.

A precanonical structure on a free \mathcal{Z}-module F consists of

  • A standard basis (t_i)_{i\in I} of F,
  • A partial order on I that is interval finite, i.e. (-\infty,i] := \{j\in I | j\leq i\} is finite for all i\in I,
  • A dualization operation, i.e. a bijection F\to F of order two that is \overline{\cdot}-semilinear and will be denoted by \overline{\cdot} as well.

If a precanonical structure is given, then one can define the \mathcal{Z}^{\pm} submodule F^{\pm} := \sum \mathcal{Z}^{\pm} t_j of F.

A canonical basis at v=0 of the precanonical structure is then a \mathcal{Z}-basis (c_i)_{i\in I} of F that satisfies:

  • \overline{c_i}=c_i and
  • c_i \in \sum_{j\leq i} \mathcal{Z}^+ t_jand c_i \equiv t_i \mod vF^+

for all i\in I. A canonical basis at v=\infty is analogously defined to be a basis (\widetilde{c}_i)_{i\in I} that satisfies

  • \overline{\widetilde{c}_i}=\widetilde{c}_i and
  • \widetilde{c}_i \in \sum_{j\leq i} \mathcal{Z}^- t_j and \widetilde{c}_i \equiv t_i \mod v^{-1}F^-

for all i\in I. The naming "at v=\infty" alludes to the fact \lim_{v\to\infty} v^{-1} =0 and hence the "specialization" v\mapsto\infty corresponds to quotienting out the relation v^{-1}=0.

One can show that there exists at most one canonical basis at v=0 (and at most one at v=\infty) for each precanonical structure. A sufficient condition for existence is that the polynomials r_{ij}\in\mathcal{Z} defined by \overline{t_j}=\sum_i r_{ij} t_i satisfy r_{ii}=1 and r_{ij}\neq 0 \implies i\leq j.

A canonical basis at v=0 (v=\infty) induces an isomorphism from \textstyle F^+\cap \overline{F^+} = \sum_i \mathbb{Z}c_i to F^+/vF^+ (\textstyle F^{-} \cap \overline{F^{-}}=\sum_i \mathbb{Z}\widetilde{c}_i \to F^{-}/v^{-1} F^{-} respectively).


Quantum groups[edit]

The canonical basis of quantum groups in the sense of Lusztig and Kashiwara are canonical basis at v=0.

Hecke algebras[edit]

Let (W,S) be a Coxeter group. The corresponding Iwahori-Hecke algebra H has the standard basis (T_w)_{w\in W}, the group is partially ordered by the Bruhat order which is interval finite and has a dualization operation defined by \overline{T_w}:=T_{w^{-1}}^{-1}. This is a precanonical structure on H that satisfies the sufficient condition above and the corresponding canonical basis of H at v=0 is the Kazhdan-Lusztig basis C_w' = \sum_{y\leq w} P_{y,w}(v^2) T_w with P_{y,w} being the Kazhdan-Lusztig polynomials.

See also[edit]