Quasi-Characters: Relaxed Notions Across Mathematics

• Quasi-characters are generalized objects that relax strict character conditions while retaining enough structural properties to facilitate classification and analysis across various mathematical domains.
• They manifest in multiple settings—from non-vanishing conditions in finite groups and orbital pairing constraints in reductive groups to recoupling-based invariant functions and modular differential equation solutions in RCFT and VOA theory.
• Their study has practical implications in representation theory, lattice gauge theory, and automorphic analysis, providing unified insights despite varied constructions and applications.

“Quasi-character” is not a single standardized notion across mathematics and mathematical physics. In current usage it can denote at least six distinct constructions: a non-vanishing condition for irreducible characters of finite groups on $p$-regular elements, a homogeneity condition for restrictions of irreducible characters to normal subgroups, a root-theoretic condition on characters and cocharacters of connected reductive groups, a basis of invariant representative functions on $G^N$ modulo diagonal conjugation, vector-valued modular functions arising from modular linear differential equations in vertex-operator and rational conformal field theory, and continuous homomorphisms from a locally compact abelian group to $\mathbf C^\times$ in automorphic settings (Paul et al., 2020, Goldring et al., 2017, Jarvis et al., 2020, Grover, 2022, Chapdelaine, 2016). Several of these usages are explicitly noted to be unrelated despite the shared terminology (Goldring et al., 2017, Youmbai et al., 2023, Mishra et al., 2022).

1. Terminological scope and recurrent patterns

The common linguistic feature of the term is that it designates an object that relaxes a stricter notion while retaining part of its structure. In finite group character theory, “quasi $p$-Steinberg” keeps only the non-vanishing on $p$-regular elements, without requiring the exact Steinberg value formula $\theta(x)=\pm |C_G(x)|_p$ (Paul et al., 2020). In the theory of reductive groups, “quasi-constant” weakens “minuscule” by allowing constant absolute pairing $m>1$ on a Weyl–Galois orbit rather than forcing values in $\{-1,0,1\}$ (Goldring et al., 2017). In RCFT and VOA theory, quasi-characters are vector-valued modular functions with integral $q$-series coefficients but without the positivity required of admissible characters (Chandra et al., 2018, Das et al., 9 Jul 2025). In the automorphic setting of $GL_2$ Eisenstein series, a quasi-character is simply a continuous homomorphism $G^N$0, with unitary characters forming a distinguished subclass (Chapdelaine, 2016).

A concise comparison is useful.

Domain	Meaning of “quasi-character”	Core relaxation
Finite groups	quasi $G^N$1-Steinberg irreducible character	non-vanishing on $G^N$2-regular elements only
Finite groups	quasi-primitive irreducible character	homogeneous restriction to every normal subgroup
Reductive groups	quasi-constant character/cocharacter	orbitwise values in $G^N$3
Compact Lie groups on $G^N$4	invariant representative function basis element	generalizes ordinary characters from $G^N$5
VOA/RCFT	VVMF with integral but not necessarily positive $G^N$6-series	drops admissibility/positivity
Automorphic $G^N$7 setting	continuous homomorphism to $G^N$8	includes non-unitary twists

This multiplicity of meanings is not accidental. Several sources explicitly warn that the same word is used differently in different subfields. The paper on quasi-constant characters states that its notion is unrelated to analytic or Harish-Chandra-style usages (Goldring et al., 2017). The note on quasi-primitive irreducible characters uses “quasi-characters” only for quasi-primitive characters in finite group theory (Youmbai et al., 2023). The classification of quasi $G^N$9-Steinberg characters of complex reflection groups likewise remarks that this usage is unrelated to quasi-characters in the Harish-Chandra or Arthur sense (Mishra et al., 2022).

2. Finite-group character theory: quasi $\mathbf C^\times$0-Steinberg and quasi-primitive characters

For a finite group $\mathbf C^\times$1 and a prime $\mathbf C^\times$2, an element $\mathbf C^\times$3 is $\mathbf C^\times$4-regular if $\mathbf C^\times$5. An irreducible character $\mathbf C^\times$6 is called quasi $\mathbf C^\times$7-Steinberg if

$\mathbf C^\times$8

This notion generalizes the classical Steinberg character by retaining only the non-vanishing on $\mathbf C^\times$9-regular elements (Paul et al., 2020). For symmetric and alternating groups, $p$0-regularity is read off from cycle type: a class is $p$1-regular iff no cycle length is divisible by $p$2 (Paul et al., 2020).

The classification for $p$3, $p$4, and their double covers is extremely rigid. For $p$5, non-linear quasi $p$6-Steinberg characters exist only for $p$7, and the complete list is:

• $p$8: $p$9 for $p$0.
• $p$1: $p$2 for $p$3; $p$4, $p$5 for $p$6.
• $p$7: $p$8, $p$9 for $\theta(x)=\pm |C_G(x)|_p$0; $\theta(x)=\pm |C_G(x)|_p$1, $\theta(x)=\pm |C_G(x)|_p$2 for $\theta(x)=\pm |C_G(x)|_p$3.
• $\theta(x)=\pm |C_G(x)|_p$4: $\theta(x)=\pm |C_G(x)|_p$5 for $\theta(x)=\pm |C_G(x)|_p$6; $\theta(x)=\pm |C_G(x)|_p$7, $\theta(x)=\pm |C_G(x)|_p$8 for $\theta(x)=\pm |C_G(x)|_p$9.
• $m>1$0: $m>1$1, $m>1$2 for $m>1$3. For $m>1$4, every non-linear irreducible character has a zero on some $m>1$5-regular class (Paul et al., 2020).

For $m>1$6, the corresponding bound is $m>1$7. The list of non-linear quasi $m>1$8-Steinberg characters is:

• $m>1$9: $\{-1,0,1\}$0 for $\{-1,0,1\}$1.
• $\{-1,0,1\}$2: $\{-1,0,1\}$3 for $\{-1,0,1\}$4; $\{-1,0,1\}$5, $\{-1,0,1\}$6 for $\{-1,0,1\}$7.
• $\{-1,0,1\}$8: $\{-1,0,1\}$9 for $q$0; $q$1 for $q$2; $q$3 for $q$4.
• $q$5: $q$6 for $q$7; $q$8 for $q$9; $GL_2$0, $GL_2$1 for $GL_2$2.
• $GL_2$3: $GL_2$4 for $GL_2$5.
• $GL_2$6: $GL_2$7 for $GL_2$8 (Paul et al., 2020).

For Schur double covers $GL_2$9 and $G^N$00, ordinary characters inflated from $G^N$01 or $G^N$02 preserve the quasi $G^N$03-Steinberg property, so the genuinely new issue is spin characters. The classification is strikingly sharp: no spin character is quasi $G^N$04-Steinberg for odd $G^N$05, while for $G^N$06 the quasi $G^N$07-Steinberg spin characters are exactly those indexed by strict partitions

$G^N$08

(Paul et al., 2020).

The same non-vanishing problem was extended to complex reflection groups $G^N$09. There the quasi $G^N$10-Steinberg condition is defined identically, and the classification reduces most cases to the symmetric-group classification plus two low-degree families coming from restriction phenomena. For $G^N$11, the only possibilities are certain $G^N$12-partitions of shape $G^N$13 corresponding to the $G^N$14 list above, together with three small mixed-shape families $G^N$15 for $G^N$16. For $G^N$17, all non-linear quasi $G^N$18-Steinberg characters arise either by restriction of those $G^N$19 cases or from two extra low-degree families: $G^N$20, $G^N$21 multiples of $G^N$22, giving quasi $G^N$23-Steinberg constituents of degree $G^N$24; and $G^N$25, $G^N$26 even, giving quasi $G^N$27-Steinberg constituents of degree $G^N$28. In particular, for $G^N$29, no non-linear quasi $G^N$30-Steinberg characters exist in $G^N$31 (Mishra et al., 2022).

A distinct finite-group usage is “quasi-primitive irreducible character.” An irreducible $G^N$32 is quasi-primitive if for every normal subgroup $G^N$33, the restriction $G^N$34 is homogeneous: $G^N$35 for some $G^N$36 and some $G^N$37 (Youmbai et al., 2023). Primitive characters are always quasi-primitive, but the converse need not hold in general (Youmbai et al., 2023). The principal counting result is orbit-theoretic: both the number of primitive irreducible characters and the number of quasi-primitive irreducible characters are divisible by $G^N$38, where $G^N$39 is the derived subgroup (Youmbai et al., 2023). The proof uses the multiplicative action of $G^N$40 on $G^N$41 and the fact that for quasi-primitive $G^N$42, the restriction $G^N$43 is irreducible, so the action is semiregular (Youmbai et al., 2023).

Methodologically, the finite-group classifications rely on explicit character formulas. For symmetric groups, the hook-length formula gives degrees and the Murnaghan–Nakayama rule supplies vanishing criteria. In the form used in the classification,

$G^N$44

and the practical consequences are that absence of an appropriate rim hook forces vanishing on a conjugacy class (Paul et al., 2020). For spin characters of double covers, Schur’s vanishing theorem and Morris recursion play the analogous role (Paul et al., 2020). This suggests that, in these settings, quasi-character phenomena are controlled less by abstract block theory than by fine combinatorics of Young and shifted diagrams.

3. Quasi-constant characters and cocharacters of reductive groups

In the theory of connected reductive groups, the relevant notion is not a class function but a weight in the root datum. Let $G^N$45 be a connected reductive group over a field $G^N$46, $G^N$47 a maximal torus, and $G^N$48 its root datum. A character $G^N$49 is quasi-constant if for every root $G^N$50 with $G^N$51 and every $G^N$52,

$G^N$53

Equivalently, for each Weyl–Galois orbit $G^N$54, the multiset of pairings $G^N$55 is contained in $G^N$56 for some integer $G^N$57 (Goldring et al., 2017). The dual definition for cocharacters exchanges roots and coroots (Goldring et al., 2017).

This notion interpolates between minuscule and cominuscule. Minuscule means $G^N$58 for all roots; quasi-constant allows a larger constant absolute value on an orbit. In the simple case, the nonzero quasi-constant characters are exactly the integer multiples of minuscule or cominuscule fundamental weights (Goldring et al., 2017). In simply-laced types, cominuscule and minuscule coincide, so quasi-constant means “multiple of a minuscule fundamental weight” (Goldring et al., 2017).

The type-by-type classification is explicit. For simple, simply connected or adjoint groups over an algebraically closed field:

• $G^N$59: all fundamental weights are minuscule, so quasi-constant characters are multiples of any $G^N$60.
• $G^N$61: quasi-constant characters are multiples of $G^N$62, $G^N$63, $G^N$64.
• $G^N$65: multiples of $G^N$66, $G^N$67.
• $G^N$68: multiples of $G^N$69.
• $G^N$70, $G^N$71, $G^N$72: no nontrivial quasi-constant characters.
• $G^N$73: multiples of $G^N$74 and $G^N$75.
• $G^N$76: multiples of $G^N$77 and $G^N$78 (Goldring et al., 2017).

The general reductive classification reduces to $G^N$79-simple factors. A character is quasi-constant iff its pullback to every $G^N$80-simple factor of the simply connected cover of $G^N$81 is quasi-constant, and on each absolutely simple factor the nontrivial components are all minuscule or all cominuscule, with a common integer scalar $G^N$82 across Galois-conjugate factors (Goldring et al., 2017).

A major application concerns Shimura varieties. For a symplectic embedding $G^N$83, the character $G^N$84 of the Hodge line bundle is quasi-constant for every such embedding (Goldring et al., 2017). When $G^N$85 is $G^N$86-simple, the positive ray of the Hodge line bundle in $G^N$87 is therefore independent of the symplectic embedding (Goldring et al., 2017). Another application is to $G^N$88-zips: if $G^N$89 is over $G^N$90 and $G^N$91 is quasi-constant, the duality construction yields a quasi-constant character $G^N$92 such that $G^N$93 is a Hasse generator for $G^N$94, implying uniform principal purity for the zip stratification at all primes $G^N$95 (Goldring et al., 2017).

The paper also formulates a canonical duality on rays. For semisimple $G^N$96, a $G^N$97-dominant quasi-constant ray in $G^N$98 determines a $G^N$99-dominant quasi-constant ray in $\mathbf C^\times$00 by replacing the unique excluded simple root in each Levi factor by the corresponding fundamental weight (Goldring et al., 2017). This suggests a structural symmetry between Hodge cocharacters and line-bundle characters that is sharper than mere root-datum duality.

4. Quasicharacters as invariant functions on $\mathbf C^\times$01

For a compact Lie group $\mathbf C^\times$02, another usage concerns invariant representative functions on $\mathbf C^\times$03 under diagonal conjugation. Let $\mathbf C^\times$04, where $\mathbf C^\times$05 is the algebra of representative functions and $\mathbf C^\times$06 acts by

$\mathbf C^\times$07

Fix irreducible unitary representations $\mathbf C^\times$08, write $\mathbf C^\times$09, and decompose the diagonal restriction $\mathbf C^\times$10 into isotypical components. A reduction scheme chooses intertwiners

$\mathbf C^\times$11

with $\mathbf C^\times$12, and defines invariant representative functions

$\mathbf C^\times$13

These are called quasicharacters (Jarvis et al., 2020).

For $\mathbf C^\times$14, quasicharacters reduce to ordinary irreducible characters $\mathbf C^\times$15 (Jarvis et al., 2020). For general $\mathbf C^\times$16, they depend on a reduction scheme, equivalently on a rooted binary tree $\mathbf C^\times$17 encoding the successive Clebsch–Gordan reductions. In the tree language,

$\mathbf C^\times$18

or, in index notation,

$\mathbf C^\times$19

Thus the quasicharacter is a contraction of $\mathbf C^\times$20 matrix elements along a fixed intertwiner pattern (Jarvis et al., 2020).

These functions form an orthogonal basis of $\mathbf C^\times$21. Their product closes in the same basis: $\mathbf C^\times$22 and the structure constants are expressed in terms of recoupling coefficients. In the tree-based formulation,

$\mathbf C^\times$23

so the multiplication law is entirely controlled by recoupling theory (Jarvis et al., 2020). The recoupling coefficients themselves factor into products over primitive $\mathbf C^\times$24-type quantities attached to the internal nodes of the tree (Jarvis et al., 2020).

For $\mathbf C^\times$25, the whole construction becomes angular momentum theory. Irreducibles are labelled by spins $\mathbf C^\times$26, multiplicities in two-fold tensor products are $\mathbf C^\times$27 or $\mathbf C^\times$28, and recouplings are Racah–Wigner coefficients. The quasicharacters become sums of diagonal matrix elements over magnetic quantum numbers, their norms are explicit in terms of $\mathbf C^\times$29, and the $\mathbf C^\times$30-symbols reduce to Wigner $\mathbf C^\times$31 symbols up to dimension factors (Jarvis et al., 2020).

The main motivation is Hamiltonian lattice gauge theory. With a maximal tree chosen in the lattice, the physical Hilbert space is $\mathbf C^\times$32, and bi-invariant operators such as Casimirs, orbit-type relations, and the Kogut–Susskind Hamiltonian can be represented in the quasicharacter basis. Multiplication by an invariant representative function $\mathbf C^\times$33 is reduced to finite-dimensional linear algebra using the structure constants $\mathbf C^\times$34, while differential operators act diagonally through Casimir eigenvalues (Jarvis et al., 2020). The paper works out explicit examples for $\mathbf C^\times$35 and $\mathbf C^\times$36, including orbit-type relations and sparse matrix elements (Jarvis et al., 2020).

Conceptually, these quasicharacters are close to spin networks. The paper states that for a single site with $\mathbf C^\times$37 incident edges, quasicharacters coincide with spin-network evaluations on a star graph reduced by the diagonal action; the difference lies in normalization conventions and in the explicit algebraic product law within $\mathbf C^\times$38 (Jarvis et al., 2020).

5. Quasi-characters in VOA and RCFT

In conformal field theory and vertex-algebra theory, “quasi-character” again has a different meaning. Here the fundamental objects are genus-one trace functions

$\mathbf C^\times$39

or Jacobi trace functions

$\mathbf C^\times$40

and quasi-characters are $\mathbf C^\times$41-series, often vector-valued, that solve modular linear differential equations, have integral Fourier coefficients after normalization, but fail positivity and therefore do not directly define characters of rational CFTs (Grover, 2022, Chandra et al., 2018).

The two-character case is the foundational example. A rank-2 vector-valued modular form has components

$\mathbf C^\times$42

with $\mathbf C^\times$43, $\mathbf C^\times$44, and Wronskian index

$\mathbf C^\times$45

For $\mathbf C^\times$46, the MLDE is the Mathur–Mukhi–Sen equation

$\mathbf C^\times$47

Quasi-characters are weight-0 vector-valued modular forms solving such MLDEs with integral coefficients but not necessarily nonnegative ones (Chandra et al., 2018, Das et al., 9 Jul 2025).

The classification of rank-2 quasi-characters at $\mathbf C^\times$48 organizes them into infinite families parameterized by

$\mathbf C^\times$49

with exponents

$\mathbf C^\times$50

and family-dependent modular $\mathbf C^\times$51-matrices that are independent of $\mathbf C^\times$52 (Das et al., 9 Jul 2025). The coefficient behavior is highly structured. For $\mathbf C^\times$53, the identity-component coefficients alternate in sign up to $\mathbf C^\times$54, with sign $\mathbf C^\times$55, and are strictly positive for all $\mathbf C^\times$56; the non-identity component is strictly positive for every $\mathbf C^\times$57. For $\mathbf C^\times$58, the identity component is strictly positive for all $\mathbf C^\times$59, while the non-identity component alternates up to $\mathbf C^\times$60, has $\mathbf C^\times$61, and is strictly negative for all $\mathbf C^\times$62 (Das et al., 9 Jul 2025). These results prove earlier conjectures about sign stabilization near $\mathbf C^\times$63 (Das et al., 9 Jul 2025), sharpening the two-character quasi-character program initiated in the 2018 classification (Chandra et al., 2018).

This sign structure is not merely descriptive. Quasi-characters form explicit bases from which admissible characters can be built by finite linear combinations within a fixed modular family: $\mathbf C^\times$64 Such combinations have

$\mathbf C^\times$65

and with suitable coefficients $\mathbf C^\times$66 can yield admissible character vectors with nonnegative integral coefficients (Das et al., 9 Jul 2025). Earlier work proved that in rank 2 all admissible characters of allowed $\mathbf C^\times$67 can be generated from quasi-characters in this way (Chandra et al., 2018).

The three-character case is substantially more intricate. A third-order MLDE

$\mathbf C^\times$68

governs $\mathbf C^\times$69 theories, and infinite families of three-character quasi-characters were conjectured and used to generate admissible characters of arbitrarily large Wronskian index (Mukhi et al., 2020). A more recent development gives a universal hypergeometric description: all $\mathbf C^\times$70 solutions can be written in terms of $\mathbf C^\times$71, taking into account monodromy at the elliptic points. Starting from known $\mathbf C^\times$72 and $\mathbf C^\times$73 solutions, the matrix-MLDE formalism produces additional basis vectors with the same multiplier; these are typically quasi-characters. Integer linear combinations of them yield new admissible $\mathbf C^\times$74, $\mathbf C^\times$75, and higher-index solutions, and the admissible points appear as integer points in a polytope (Govindarajan et al., 28 Oct 2025).

In affine $\mathbf C^\times$76 current algebra at admissible fractional levels, quasi-characters arise from unflavoured even characters. The paper on fractional levels shows that outside three special classes—threshold levels, positive half-odd integer levels, and the isolated level $\mathbf C^\times$77—the resulting vectors are quasi-characters: vector-valued modular functions with integer $\mathbf C^\times$78-series coefficients violating positivity (Grover, 2022). At half-odd integer levels, the even characters map to differences of $\mathbf C^\times$79 characters, which explains why they do not define RCFTs despite often having positive coefficients to very high order (Grover, 2022).

The VOA perspective places these modular phenomena in a geometric framework. For a conformal vertex algebra $\mathbf C^\times$80, quasi-lisse and stably quasi-lisse conditions imply holonomicity of the sheaf of charged conformal blocks over the moduli of elliptic curves with line bundles. Under stable rationality, the space of flat sections is spanned by trace functions on irreducible stable modules (Arakawa et al., 28 May 2026). The Jacobi-invariant connection satisfies Ward identities

$\mathbf C^\times$81

$\mathbf C^\times$82

and the resulting flat sections transform under the Jacobi group as vector-valued Jacobi forms (Arakawa et al., 28 May 2026). In particular, for admissible affine vertex algebras $\mathbf C^\times$83, the dimension of the space of conformal blocks equals the number of admissible weights at level $\mathbf C^\times$84, and the charged trace functions form a vector-valued Jacobi form of weight $\mathbf C^\times$85 and index

$\mathbf C^\times$86

(Arakawa et al., 28 May 2026). This gives a rigorous modular-geometric explanation for the appearance of quasi-character solutions in MLDE classifications.

6. Automorphic quasi-characters and broader perspective

In the automorphic setting of real-analytic Eisenstein series, a quasi-character is simply a continuous homomorphism from a locally compact abelian group to $\mathbf C^\times$87. For a locally compact abelian group $\mathbf C^\times$88,

$\mathbf C^\times$89

is the group of quasi-characters, while characters are those whose image lies in $\mathbf C^\times$90 (Chapdelaine, 2016). For $\mathbf C^\times$91 a totally real field and $\mathbf C^\times$92, the monograph isolates a lattice of integral unitary characters

$\mathbf C^\times$93

and, relative to a finite-index subgroup $\mathbf C^\times$94, defines $\mathbf C^\times$95-integral quasi-characters

$\mathbf C^\times$96

where $\mathbf C^\times$97, $\mathbf C^\times$98, and $\mathbf C^\times$99 is built from logarithms of a $p$00-basis of $p$01 (Chapdelaine, 2016). These are the monograph’s “multiplicative integral quasi-characters.”

They enter directly into the definition of twisted $p$02 real-analytic Eisenstein series. Given lattices $p$03, a parameter matrix

$p$04

an integral weight $p$05, and $p$06, the Eisenstein series

$p$07

is defined by a lattice sum twisted by $p$08, $p$09, additive characters from $p$10, and the analytic factor $p$11 (Chapdelaine, 2016). The parameter matrix is acted on by $p$12 through an “upper right action,” and the associated Cartan involution $p$13 controls the functional equation (Chapdelaine, 2016).

The completed series satisfies a functional equation of the form

$p$14

where $p$15 (Chapdelaine, 2016). The Fourier expansion is explicit, with constant terms expressed in terms of partial zeta functions twisted by $p$16, and more general $p$17-integral quasi-characters $p$18 can be incorporated formally in the same framework (Chapdelaine, 2016).

Across all these subjects, the term “quasi-character” therefore functions as a marker of controlled generalization rather than a univocal definition. In finite groups it isolates non-vanishing or homogeneous-restriction phenomena; in reductive groups it encodes orbitwise rigidity of root pairings; in compact-group invariant theory it names a recoupling-adapted basis of gauge-invariant functions; in RCFT and VOA theory it identifies MLDE solutions that preserve modularity and integrality but not positivity; and in automorphic analysis it retains its classical meaning of a continuous homomorphism to $p$19. A plausible implication is that the persistence of the prefix “quasi-” reflects a shared methodological pattern: the relaxation of one axiom while preserving enough structure to retain classification, analytic continuation, or representation-theoretic control.