Canonical Basis Character Formula

• Canonical Basis Character Formula is a representation-theoretic framework that expresses module characters using distinguished bases defined by bar invariance and triangularity.
• It provides an explicit computational mechanism by translating standard basis elements into canonical or dual canonical bases via precise coefficient transition formulas.
• The formula underpins methodologies in highest-weight, modular, superalgebra, and quantum symmetric pair settings, enabling accurate decomposition and multiplicity analyses.

A canonical basis character formula is a representation-theoretic statement that identifies characters, multiplicities, or Grothendieck-group classes of distinguished modules with coefficients of a canonical, dual canonical, $\imath$-canonical, or $p$-canonical basis in a quantum-group or Hecke-theoretic model. Across the literature, the common pattern is the conjunction of bar-invariance and triangularity: one fixes a standard basis $m$, seeks a distinguished basis element $b$ satisfying $\overline{b}=b$ and $b=m+\text{lower terms}$, and then interprets the transition coefficients as character data. In some works the formula is an explicit expansion of simples in terms of standards; in others it is a computational algorithm for canonical basis elements; and in still others it appears as a basis-like inversion or refined character formula whose role is analogous rather than identical to Lusztig’s canonical-basis formalism (Antor, 2023, Riche et al., 2015, Cheng et al., 2 Mar 2026).

1. Defining paradigm: bar invariance, triangularity, and transition coefficients

The basic canonical-basis mechanism begins with a standard basis and a bar involution. In the positive part $U_v^+(\mathfrak g)$ of a quantized enveloping algebra of finite ADE type, canonical basis elements are characterized by bar-invariance,

$\overline{b}=b,$

and triangularity with respect to a standard monomial basis,

$b=m+\sum_{m'<m} c_{m'}\,m'.$

This is the framework used to compute canonical basis elements from monomial data in finite ADE type (Antor, 2023).

The same paradigm appears in Fock-space models. For the Leclerc–Thibon canonical basis $\{G(\lambda)\}$ of the Fock space representation of $p$0, the defining conditions are

$p$1

and the coefficients

$p$2

are the $p$3-decomposition numbers (Chuang et al., 2015). In higher-level Fock spaces for $p$4, the canonical basis $p$5 attached to standard symbols is characterized by

$p$6

with coefficients $p$7 supported on blocks having the same multiset of entries (Jacon et al., 23 Jan 2026).

In categorical character formulas the same structure is transported to Grothendieck groups. For finite $p$8-superalgebras of type $p$9, there is a $m$0-linear isomorphism

$m$1

such that

$m$2

where $m$3 is a standard basis element and $m$4 is Lusztig’s dual canonical basis element. The dual canonical basis is itself characterized by bar-invariance and triangularity: $m$5 Thus the “character formula” is literally a basis-identification between simple-module classes and dual canonical basis vectors (Cheng et al., 2 Mar 2026).

This pattern extends to the queer Lie superalgebra $m$6. In the half-integer-weight finite-dimensional category, the character formula identifies irreducible characters with the dual canonical basis of a type $m$7 $m$8-wedge space at $m$9: $b$0 Equivalently,

$b$1

Here the coefficients arise from the transition matrices between standard, canonical, and dual canonical bases (Cheng et al., 2015).

2. Canonical basis as an explicit computational object

A canonical basis character formula is not only a structural identification; it is also a computational problem. One major development is the replacement of geometric or recursive input by explicit algebraic formulas for pairings and monomials. For finite ADE quantum groups, a new closed expression for the standard bilinear form $b$2 on monomials turns canonical basis computation into a Lusztig–Shoji style triangular elimination problem. If $b$3 is an ordered monomial basis and $b$4 the canonical basis, then one seeks

$b$5

and the explicitly known pairing matrix

$b$6

replaces the usual indirect input. In type $b$7, the same algorithm computes composition multiplicities of standard modules for the affine Hecke algebra of $b$8, and the method is further extended to compute dimensions of simple modules (Antor, 2023).

Hall-algebra methods furnish another computational route. For the positive part of the quantum loop algebra $b$9, distinguished words determine monomials $\overline{b}=b$0 with triangular expansion in the PBW/Hall basis,

$\overline{b}=b$1

The canonical basis element $\overline{b}=b$2 is then the unique bar-invariant element

$\overline{b}=b$3

obtained by recursively correcting the monomial basis. This yields an explicit algorithm for canonical basis computation and concrete formulas for slices of the basis of $\overline{b}=b$4, including the part associated with modules of Loewy length at most $\overline{b}=b$5 (Du et al., 2016).

Affine type $\overline{b}=b$6 crystals supply nonrecursive formulas in restricted regions of the crystal graph. On finite faces generated by one or two residues, the canonical basis element attached to a multipartition can be written without recursion. For $\overline{b}=b$7, the shape polynomial is the Gaussian binomial coefficient

$\overline{b}=b$8

with recurrence

$\overline{b}=b$9

For $b=m+\text{lower terms}$0, the coefficients are described by inversion statistics of binary words through explicit exponent formulas such as

$b=m+\text{lower terms}$1

These results give non-recursive canonical basis formulas on finite faces of the block-reduced crystal $b=m+\text{lower terms}$2 (Amara-Omari et al., 2023).

A related nonrecursive theory exists for symmetric Kashiwara crystals of type $b=m+\text{lower terms}$3 and rank $b=m+\text{lower terms}$4. For $b=m+\text{lower terms}$5, the paper constructs explicit canonical basis elements for external weights of defects $b=m+\text{lower terms}$6 and $b=m+\text{lower terms}$7, with coefficients governed by inversion numbers of binary choice sequences. In the top-row case,

$b=m+\text{lower terms}$8

and analogous formulas hold for the next external layer (Amara-Omari et al., 2020).

Higher-level Fock spaces also admit closed formulas in special regimes. The generalization of the Leclerc–Miyachi formula to arbitrary level yields

$b=m+\text{lower terms}$9

for standard symbols satisfying the compatibility condition $U_v^+(\mathfrak g)$0, while ordered symbols satisfy

$U_v^+(\mathfrak g)$1

A column removal theorem and monomiality results for $U_v^+(\mathfrak g)$2 further reduce computation of canonical basis elements in $U_v^+(\mathfrak g)$3-Fock spaces (Jacon et al., 23 Jan 2026).

3. Character formulas in highest-weight and modular representation theory

In highest-weight and modular settings, the phrase “canonical basis character formula” often means that module characters are read off from canonical-basis coefficients after an appropriate specialization. For reductive algebraic groups in characteristic $U_v^+(\mathfrak g)$4, the principal block $U_v^+(\mathfrak g)$5 is conjecturally governed by the diagrammatic Hecke category of the affine Weyl group, and this implies character formulas for simple and tilting modules in terms of the anti-spherical $U_v^+(\mathfrak g)$6-canonical basis. The indicated formulas are

$U_v^+(\mathfrak g)$7

for tilting modules and

$U_v^+(\mathfrak g)$8

schematically for simple modules. For $U_v^+(\mathfrak g)$9, the conjectural action is proved via $\overline{b}=b,$0-Kac–Moody actions, so the $\overline{b}=b,$1-canonical basis character formulas hold in that case (Riche et al., 2015).

The comparison between canonical bases and decomposition numbers is especially explicit for Fock spaces and Schur algebras. Under restriction from $\overline{b}=b,$2 to parabolic subalgebras

$\overline{b}=b,$3

the Fock space decomposes as

$\overline{b}=b,$4

and each $\overline{b}=b,$5 is a based module carrying its own canonical basis. The comparison theorem states

$\overline{b}=b,$6

so the same canonical basis coefficients govern both the ambient and restricted modules. In the cases $\overline{b}=b,$7 and $\overline{b}=b,$8, these coefficients specialize at $\overline{b}=b,$9 to decomposition numbers of Schur algebras: $b=m+\sum_{m'<m} c_{m'}\,m'.$0 Runner-removal factorization gives product formulas for the coefficients (Chuang et al., 2015).

The spin representation theory of symmetric groups furnishes a further example. In the level-$b=m+\sum_{m'<m} c_{m'}\,m'.$1 $b=m+\sum_{m'<m} c_{m'}\,m'.$2-deformed Fock space of type $b=m+\sum_{m'<m} c_{m'}\,m'.$3, canonical basis vectors

$b=m+\sum_{m'<m} c_{m'}\,m'.$4

have coefficients $b=m+\sum_{m'<m} c_{m'}\,m'.$5, the “$b=m+\sum_{m'<m} c_{m'}\,m'.$6-decomposition numbers.” For bar-weight $b=m+\sum_{m'<m} c_{m'}\,m'.$7 and $b=m+\sum_{m'<m} c_{m'}\,m'.$8 blocks, a Richards-style combinatorial formula determines these coefficients explicitly. In weight $b=m+\sum_{m'<m} c_{m'}\,m'.$9,

$\{G(\lambda)\}$0

while in weight $\{G(\lambda)\}$1 the formulas involve the $\{G(\lambda)\}$2-statistic, colors, dominance intervals, and exceptional partitions $\{G(\lambda)\}$3. The specialization at $\{G(\lambda)\}$4 is expected to recover reduced spin decomposition numbers in the corresponding defect-$\{G(\lambda)\}$5 blocks (Fayers, 2019).

These examples show that the canonical basis character formula is often a decomposition-number formula in disguise: the basis coefficients, sometimes after evaluation at $\{G(\lambda)\}$6, encode standard-filtration multiplicities, simple characters, or block decomposition matrices.

4. Super and $\{G(\lambda)\}$7-superalgebra formulations

For Lie superalgebras and finite $\{G(\lambda)\}$8-superalgebras, canonical basis character formulas appear as direct analogues of Kazhdan–Lusztig theory, but with tensor product models adapted to super phenomena. The finite $\{G(\lambda)\}$9-superalgebra of type $p$00, denoted $p$01, admits parabolic BGG-type categories $p$02 whose Grothendieck groups categorify tensor product modules of irreducible polynomial representations and their duals over $p$03. Standard modules $p$04 correspond to standard basis vectors $p$05, irreducibles $p$06 correspond to Lusztig’s dual canonical basis elements $p$07, and the basis-identification

$p$08

gives the character formula. The finite-dimensional category $p$09 is a special case: $p$10 The theorem is presented as a uniform generalization of several earlier character formulas in BGG categories for Lie superalgebras and $p$11-algebras of type $p$12 (Cheng et al., 2 Mar 2026).

For $p$13, two quantum-group types occur. In Brundan’s original formulation for integer weights, the conjectural character formula is expressed in terms of a type $p$14 canonical basis on tensor space: $p$15 For half-integer weights, the relevant structure changes to type $p$16. The tensor space $p$17 and the type $p$18 $p$19-wedge space $p$20 carry bar involutions defined by the quasi-$p$21-matrix, with canonical and dual canonical bases

$p$22

The finite-dimensional half-integer character formula is then

$p$23

which identifies the irreducible character with the specialized dual canonical basis element (Cheng et al., 2015).

A different, but still canonical-basis-type, superalgebra phenomenon appears in the categorification of the generic Su–Zhang character formula for $p$24. For a dominant integral $p$25-generic weight $p$26, the simple character is

$p$27

The categorifying object is the narrow Verma module

$p$28

whose character is the modified Verma character

$p$29

A BGG-type resolution by narrow Verma modules then categorifies the finite alternating-sum character formula. This is not a canonical basis theorem in Lusztig’s sense, but it shares the same structural role of converting an alternating expansion of standard objects into an exact categorical character formula (Hirota, 18 Dec 2025).

5. Quantum symmetric pairs and $p$30-canonical bases

Quantum symmetric pairs replace the usual quantum group by a coideal subalgebra $p$31, and the corresponding basis theory is governed by a twisted bar involution. For a quantum symmetric pair $p$32 of arbitrary finite type, the $p$33-canonical basis on a based $p$34-module $p$35 is the unique basis

$p$36

such that

$p$37

The bar involution is defined using the intertwiner $p$38,

$p$39

Finite-dimensional simple $p$40-modules, their tensor products, and the modified form $p$41 all admit such $p$42-canonical bases (Bao et al., 2016).

The triangular expansion

$p$43

is the $p$44-analogue of a canonical basis character formula: it expresses the new basis adapted to the symmetric pair in terms of the ordinary canonical basis. The associated bilinear form makes the basis almost orthonormal,

$p$45

which plays the same structural role as orthonormality in Lusztig’s characterization (Bao et al., 2016).

In rank one, the theory becomes completely explicit. For Letzter’s coideal subalgebra $p$46 with

$p$47

the modified form splits into even and odd pieces, each isomorphic to $p$48. The $p$49-canonical basis elements have closed shifted-product formulas: $p$50

$p$51

The geometric basis from [LW18] and the algebraic basis from [BW18]/[BeW18] are shown to coincide, so in this rank-one setting the canonical basis characters can be read off directly from these explicit polynomials in $p$52 (Li, 2019).

This rank-one case clarifies a frequent misconception. A canonical basis character formula need not always be an elaborate Kazhdan–Lusztig polynomial identity; in some settings it can be an explicit closed polynomial formula for basis vectors themselves. The quantum symmetric pair literature shows both extremes: an abstract basis theorem in arbitrary finite type and a concrete shifted-product formula in rank one (Bao et al., 2016, Li, 2019).

6. Related refinements, analogues, and boundary cases

Several works use “canonical” basis language or basis-like expansions in ways that are adjacent to, but not identical with, the Lusztig–Kashiwara character-formula paradigm. These cases broaden the semantic range of the term while preserving its emphasis on distinguished expansions and structural coefficients.

One analogue is the canonical Brauer induction formula. For a finite group $p$53, the canonical section

$p$54

of the induction map yields a distinguished expansion

$p$55

for any character $p$56. The derived invariant

$p$57

is shown to equal a multiplicity in a virtual character built from Adams operations,

$p$58

and satisfies a non-negativity and eigenvalue criterion. The paper explicitly describes this as a canonical “basis formula” viewpoint for characters, though the basis here consists of induced linear characters rather than canonical basis vectors in a quantum-group module (Boltje et al., 3 Oct 2025).

Another analogue occurs in the trivial source algebra. The primitive idempotents $p$59 of $p$60 are expressed as linear combinations of the canonical $p$61-basis $p$62: $p$63 with coefficients involving irreducible Brauer characters and Möbius functions of fixed-point posets. This is an inversion formula rather than a canonical-basis character formula in the quantum-group sense, but it exemplifies the same principle that a distinguished integral basis can encode character-theoretic data through an explicit change-of-basis matrix (Barker, 2018).

Refined Weyl character formulas provide a different boundary case. For $p$64, the symmetric power $p$65 admits a canonical refined weight-space decomposition

$p$66

as comodules for the schematic normalizer $p$67 of the diagonal torus. The refined character formula is

$p$68

for paired Weyl-orbit pieces, with the even-degree middle term $p$69 in degree $p$70. Summing these refined characters recovers the classical Weyl character formula. This is not a canonical basis formula, but it is a canonical decomposition of a character into distinguished Weyl-orbit summands (Maakestad, 2024).

Finally, canonical basis methods can govern formulas beyond characters in the narrow sense. For simple Lie algebras, Lusztig’s theory singles out a canonical Chevalley basis unique up to global sign, and the structure constants are given explicitly by

$p$71

This is a canonical structure-constant formula rather than a module-character formula, but it shows the same phenomenon: canonical basis normalization converts ambiguity of signs and coefficients into explicit combinatorial data (Geck et al., 2024).

Taken together, these variants suggest that “canonical basis character formula” has both a strict and an extended usage. In the strict usage, it refers to the identification of simple, standard, tilting, or irreducible-module characters with canonical-basis transition coefficients in quantum-group or Hecke-category models. In the extended usage, it refers to any canonical basis or basis-like expansion whose coefficients encode character-theoretic, multiplicity-theoretic, or structural information in a distinguished and functorial way.