Canonical Bases in Higher-Level Fock Spaces

• Canonical basis elements are explicit bar-invariant bases that bridge combinatorics, categorification, and modular representation theory.
• The construction employs q-wedge products, abacus models, and straightening rules to index multipartitions and compute q-decomposition matrices.
• Recent advances provide closed formulas, reduction theorems, and categorified interpretations via quiver Schur algebras, enhancing practical computations.

Canonical basis elements in higher-level Fock spaces play a central role in the representation theory of quantum groups, Hecke algebras, and related categorifications. They provide explicit, bar-invariant bases for modules arising from combinatorial and geometric constructions, encode graded decomposition numbers for cyclotomic Schur algebras, and serve as bridges between combinatorics, categorification, and modular representation theory. In higher-level settings, their structure is governed by multipartitions and symbols, straightening relations for q-wedges, and intricate conditions imposed by multicharges. Recent advances enable closed formulas, reduction theorems, and categorified interpretations via quiver Schur algebras.

1. Higher-Level q-Deformed Fock Spaces and Canonical Basis Construction

Given integers $n\geq 2$ and level $\ell\geq 1$, together with a multi-charge $$\mathbf{s} = (s_1, \ldots, s_\ell)\in\mathbb{Z}^\ell$$, the level-$\ell$ q-deformed Fock space $F_q[\mathbf{s}]$ is a $\mathbb{Q}(q)$-vector space with standard basis labelled by multipartitions $$\lambda = (\lambda^{(1)}, \ldots, \lambda^{(\ell)})$$, each $\lambda^{(j)}$ being a partition. This basis may be realized through ordered q-wedge products $U_{k_1}\wedge \cdots \wedge U_{k_r}$, subject to straightening rules that depend on $(n,\ell)$ (Iijima, 2011, Iijima, 2012). The bijection between q-wedges and multipartitions is mediated via the abacus model and Maya diagrams: the combinatorics of bead configurations encodes module elements and allows explicit indexing by "symbols" representing multipartitions (Jacon et al., 23 Jan 2026).

The action of the quantum group $U_q(\mathfrak{gl}_\infty)$, or $U_q(\widehat{\mathfrak{gl}_n})$, is defined via Chevalley generators and Heisenberg operators, where addition/removal of $i$-residue nodes in the Young diagrams corresponds to raising/lowering operators on $F_q[\mathbf{s}]$. The bar-involution, central to canonical basis theory, sends $q\mapsto q^{-1}$ and reverses wedge order up to explicit powers of $q$, fixing generators (Iijima, 2011, Chuang et al., 2015). The involution admits a unitriangular expansion in the dominance (or Jantzen) order imposed on multipartitions.

2. Definition and Properties of Canonical Basis Elements

Canonical basis elements in higher-level Fock spaces, as constructed by Uglov and Leclerc-Thibon, are characterized as unique bar-invariant elements $G^{\pm}(\lambda;\mathbf{s})$ satisfying

$$\overline{G^{\pm}(\lambda;\mathbf{s})} = G^{\pm}(\lambda;\mathbf{s}),$$

$$G^{+}(\lambda;\mathbf{s}) = |\lambda;\mathbf{s}\rangle + q\cdot \mathbb{Z}[q]\text{-combination of lower }|\mu;\mathbf{s}\rangle,$$

$$G^{-}(\lambda;\mathbf{s}) = |\lambda;\mathbf{s}\rangle + q^{-1}\cdot \mathbb{Z}[q^{-1}]\text{-combination of lower }|\mu;\mathbf{s}\rangle,$$

for multipartitions $\lambda$ in the dominance order (Iijima, 2011, Iijima, 2012). The transition matrices expressing the canonical basis in terms of the standard basis — the q-decomposition matrices $\Delta^\pm_{\lambda,\mu}(q)$ — encode graded decomposition numbers for associated Schur algebras, rational Cherednik algebra category $\mathcal{O}$, and Ariki-Koike algebras (Stroppel et al., 2011, Jacon et al., 23 Jan 2026).

The canonical basis can also be indexed combinatorially using symbols, with explicit closed formulas now available under various structural hypotheses. In particular, for standard symbols with multi-charge $\mathbf{v}$, there is a monomial expression for $G(S)$ and a generalization of the Leclerc-Miyachi formula for arbitrary level: $$G(S) = \sum_{\sigma \in \tilde{\mathrm{Sym}}(S)} q^{\ell(\sigma)} S^\sigma,$$ where $\tilde{\mathrm{Sym}}(S)$ are spine-permutations and $\ell(\sigma)$ is the Coxeter length (Jacon et al., 23 Jan 2026).

3. Reduction and Comparison Theorems: Lowering Level via Multi-charge Separation

Two pivotal results describe stability of canonical basis elements under changes in level determined by extremal multi-charges (Iijima, 2011):

• Theorem A: If $s_j$ is sufficiently large for $|\lambda;\mathbf{s}\rangle$ (in the sense that $\lambda^{(j)}=\emptyset$ and $s_j-s_i\geq|\lambda|$ for all $i\neq j$), then the corresponding block of the q-decomposition matrix at level $\ell$ coincides with the one for level $\ell-1$ with the $j$-th coordinate removed:

$$\Delta^\pm_{\lambda,\mu}(q) = \Delta^\pm_{\hat{\lambda},\hat{\mu}}(q).$$

• Theorem B: If $s_j$ is sufficiently small for $|\mu;\mathbf{s}\rangle$ and $\mu^{(j)}=\emptyset$, then the same identification holds.

These results use abacus combinatorics and wedge straightening rules, and enable inductive or truncation-based computations for canonical bases as the level increases or decreases, thereby explicating “stability” phenomena in decomposition matrices (Iijima, 2011).

4. Closed Formulas, Algorithms, and Inversion Relations

Explicit formulas for canonical basis elements are available in several situations:

• The generalized Leclerc-Miyachi theorem expresses $G(S)$ for standard symbols via spine-permutations, producing monomial expansions (Jacon et al., 23 Jan 2026).
• In the asymptotic multi-charge regime where $v_j-v_{j+1}\gg0$, canonical bases factor as tensor products:

$G(S) = G(S_1)\otimes \cdots \otimes G(S_l),$

with each $S_j$ a smaller charge symbol (Jacon et al., 23 Jan 2026).

• At level 3, all canonical basis elements are single monomials (Jacon et al., 23 Jan 2026).

For completion Fock spaces with varying sequence $b$, Cao–Lam provide symmetric bilinear forms with canonical and dual canonical bases in perfect duality, as well as explicit inversion formulas for the standard basis in terms of the canonical/dual canonical bases: $$M_f = \sum_{h\leq_b f} t_{f,h}(q^{-1}) L_b(h) = \sum_{h\leq_b f} \ell_{f,h}(q^{-1}) T_b(h)$$ alongside Brundan’s recursive algorithm for computation and stepwise isomorphisms for arbitrary $b$ (Cao et al., 2015).

5. Categorification and Relation to Quiver Schur Algebras

Canonical basis elements in higher-level Fock spaces correspond, under categorification, to indecomposable projective modules in graded cyclotomic and quiver Schur algebras. The graded decomposition numbers $d_{\xi,\eta}(q)$ coincide with transition coefficients between canonical and standard bases: $$b_\xi = u_\xi + \sum_{\eta<\xi} d_{\xi,\eta}(q) u_\eta,$$ and under the bar-involution and Serre-twisted duality, they remain fixed, confirming their canonical status (Stroppel et al., 2011).

Categorification yields weak module categorifications for $U_q(\widehat{\mathfrak{gl}_n})$-modules, with the actions of $F_i$, $E_i$, $K_i$ represented geometrically (convolution, induction, restriction) and diagrammatically (splits, merges, dots). The decomposition matrices for Ariki-Koike algebras and rational Cherednik category $\mathcal{O}$ can be recovered from these canonical bases (Stroppel et al., 2011, Iijima, 2012, Jacon et al., 23 Jan 2026).

6. Representation-Theoretic and Combinatorial Applications

The canonical basis in higher-level Fock spaces provides graded decomposition numbers for cyclotomic Hecke and Schur algebras, realizes Calogero-Moser cellular characters in complex reflection group types such as $G(\ell,1,n)$, and encodes branching coefficients under restriction to $U_q(\mathfrak{sl}_n)$: $$G(\boldsymbol{\lambda}) \equiv |\boldsymbol{\lambda}\rangle + \sum_{\boldsymbol{\mu}<\boldsymbol{\lambda}} d_{\mathbf{\mu}}^{\mathbf{\lambda}}(q) |\boldsymbol{\mu}\rangle$$ with factorization theorems stating that decompositions across components having non-adjacent residues factor as products, vanishing otherwise (Chuang et al., 2015).

Structure theorems such as the column removal theorem enable reduction of complex computations to lower charge cases, simplifying both combinatorial and categorical analyses (Jacon et al., 23 Jan 2026). The product formula (higher-level Leclerc-Thibon theorem) formally parallels the Steinberg tensor product theorem, decomposing canonical basis elements into Schur functor actions on restricted and twisted parts (Iijima, 2012).

7. Examples and Explicit Computations

Worked examples, such as for $n=\ell=2$, $s=(3,-3)$, and various choices of multipartitions, demonstrate direct computations of $q$-decomposition numbers: $$\Delta^-_{(∅,(6)),(∅,(5,1));(3,-3)}(q)=\Delta^-_{(6),(5,1);(-3)}(q) = -q^{-1}$$ and analogous results under level reduction (Iijima, 2011, Iijima, 2012).

In small ranks, canonical basis elements admit explicit expansions and correspond to projective resolutions:

• For the cyclotomic $q$-Schur algebra at $n=2$, the expansion $b_{(1,1)} = u_{(1,1)} + q \cdot u_{(2)}$ directly matches graded decomposition numbers (Stroppel et al., 2011).
• In the general symbol framework, monomial expansions of canonical basis elements are algorithmically computable and encode the block structure and cellular character data for associated representation categories (Jacon et al., 23 Jan 2026).

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References:

(Iijima, 2011, Stroppel et al., 2011, Iijima, 2012, Chuang et al., 2015, Cao et al., 2015, Jacon et al., 23 Jan 2026)