Dual Canonical Basis

• Dual Canonical Basis is a distinguished Q(q)-basis in quantum group theory, characterized by bar-invariance and a triangularity property relative to the dual PBW basis.
• It is constructed through dualizing quantum nilpotent subalgebras using Lusztig's bilinear form and quantum twist maps, ensuring reverse lexicographic unitriangular expansion.
• Its structural properties support applications in total positivity, quantum cluster algebras, and categorification, as exemplified by explicit constructions in finite types like 𝖘𝖑₃.

A dual canonical basis is a distinguished $\mathbb{Q}(q)$-basis appearing in the representation theory of quantum groups, especially within quantum nilpotent subalgebras, quantum unipotent cells, coordinate algebras of algebraic groups, and related structures. This basis is uniquely characterized by bar-invariance, a triangularity property with respect to the dual PBW (Poincaré–Birkhoff–Witt) basis, and compatibility with the quantum group structure, and it plays a central role in the study of total positivity, quantum cluster algebras, categorification, and the theory of quantum symmetric pairs.

1. Construction in Quantum Nilpotent Subalgebras

Let $\mathfrak{g}$ be a symmetrizable Kac–Moody Lie algebra and $U_q = U_q(\mathfrak{g})$ its quantized enveloping algebra. Fix a Weyl group element $w\in W$ (with reduced decomposition $w = s_{i_1}\cdots s_{i_\ell}$ and multi-index $i = (i_1,\ldots,i_\ell)$). The quantum nilpotent subalgebra associated to $w$ is defined as

$U_q^-(w) = U_q^- \cap T_w(U_q^-)$

where $U_q^-$ is the negative part of $U_q$ generated by $f_i$, and $T_w = T_{i_1}\cdots T_{i_\ell}$ is the composition of Lusztig’s braid group automorphisms.

A PBW-type basis of $U_q^-(w)$ is constructed by

$$F(a,i) = T_{i_1}\cdots T_{i_{k-1}}(f_{i_k})^{a_k} \cdots T_{i_1}(f_{i_2})^{a_2} f_{i_1}^{a_1}$$

for $a=(a_1,\ldots,a_\ell)\in\mathbb{Z}_{\geq0}^\ell$. Dualizing with respect to Lusztig’s bilinear form $(\cdot,\cdot)_L$ gives the dual PBW basis elements

$F^*(a, i) = F(a,i) / (F(a,i),F(a,i))_L$

The dual canonical basis in $U_q^-$, denoted $B^{up} = \{G^{up}(b) \mid b \in B(\infty)\}$, is characterized by bar-invariance under Lusztig's dual-bar involution $\omega$, and a triangularity property with respect to the dual PBW basis: $$G^{up}(b) \in F^*(a,i) + \sum_{a'<a} q\,\mathbb{Z}[q]\,F^*(a', i)$$ for suitable $a = (a_1,\ldots,a_\ell)$. The set $U_q^-(w) \cap B^{up}$ is a $Q(q)$-basis of $U_q^-(w)$ and the basis elements are uniquely determined by leading PBW index $a$ and bar-invariance (Kimura et al., 2016).

2. Quantum Twist Maps and Bijections

Quantum twist maps $\Theta_w$ (due to Lenagan–Yakimov) are defined via

$\Theta_w = T_w \circ S \circ \omega$

where $S$ is the antipode, and $\omega$ is the Chevalley involution of $U_q$. They satisfy $\Theta_w(U_q^-(w)) = U_q^-(w^{-1})$ and, crucially, the dual canonical basis is permuted under $\Theta_{w^{-1}}$: $$\Theta_{w^{-1}}(G^{up}(b(a,i))) = G^{up}(b(a_{rev}, i_{rev}))$$ where $a_{rev}$ and $i_{rev}$ denote reversal of the index sequences (Kimura et al., 2016).

3. Triangularity and Unipotent Quantum Minors

Expand any dual PBW basis element in terms of the dual canonical basis: $$F^*(a, i) = \sum_{a'} c_{a, a'}(q)\,G^{up}(b(a',i))$$ The coefficients $c_{a, a'}(q)$ are $0$ unless $a' <_{rev} a$ (reverse lex order), $c_{a, a}(q)=1$, and $c_{a, a'}(q)\in q\mathbb{Z}[q]$ for $a'<_{rev} a$. Thus, the transition matrix from the dual PBW basis to the dual canonical basis is upper unitriangular in the reverse lexicographic order (Kimura et al., 2016).

Unipotent quantum minors $D_{u, w}^-(\lambda)$, defined using extremal vectors in highest/lowest weight modules, always lie in the dual canonical basis. Under the twist $\Theta_{w^{-1}}$, these minors permute compatibly: $$\Theta_{w^{-1}}(D_{u, w'}^-(\lambda)) = D_{w^{-1}w', w^{-1}u}^-(\lambda)$$ when $u, w' \leq w$ in the Bruhat order (Kimura et al., 2016).

4. Categorification and Quantum Cluster Algebras

The dual canonical basis is compatible with the quantum cluster algebra structure on $U_q^-(w)$. Specifically, the quantum cluster variables (and thus all quantum cluster monomials) coincide with explicit elements of the dual canonical basis up to scalar powers of $q$. The twist symmetries are compatible with this structure (Kimura et al., 2016).

From the viewpoint of categorification, this triangularity manifests as the filtration of standard modules by proper standard modules, for instance in quiver Hecke categorification (McNamara). In finite type, reverse-unitriangularity can be deduced from crystal-theoretic arguments.

5. Explicit Example: $\mathfrak{g} = \mathfrak{sl}_3$, $w=s_1s_2s_1$

For the longest element $w=s_1s_2s_1$:

• The PBW generators for $a = (a_1, a_2, a_3)$ are $$F(a,i) = f_1^{a_1} T_1(f_2)^{a_2} T_1T_2(f_1)^{a_3}$$.
• In $\mathfrak{sl}_3$, up to $q$-commutation relations: $$f_1^{a_1} (f_2 f_1 - q^{-1} f_1 f_2)^{a_2} f_2^{a_3}$$.
• The twist map exchanges $f_1 = T_1T_2(f_1)$ with $T_1(f_2)$ and, in general, $\Theta_{w^{-1}}$ carries dual canonical basis elements labeled by $a$ to those labeled by $a_{rev}$.
• This example concretely exhibits the general triangularity and twist-induced bijection between bases (Kimura et al., 2016).

6. Summary Table: Structural Properties

Structure	Property/Characterization	Reference
Dual canonical basis	Unique bar-invariant basis with PBW-unitriangularity	(Kimura et al., 2016)
Quantum twist map	Bijection between dual canonical bases, reverses PBW indices	(Kimura et al., 2016)
Expansion of dual PBW basis	Upper unitriangular transition matrix in reverse lex order	(Kimura et al., 2016)
Unipotent quantum minors	Lie in dual canonical basis, permuted by twist	(Kimura et al., 2016)
Quantum cluster variables	Elements of dual canonical basis (up to $q$-power)	(Kimura et al., 2016)

The dual canonical basis thus provides a unifying algebraic and combinatorial structure within the representation theory of quantum groups and their subalgebras, encoding deep compatibility with cluster algebra structures, categorification, and symmetry via quantum twists.