--- title: Canonical Basis Character Formula url: https://www.emergentmind.com/topics/canonical-basis-character-formula type: topic --- # Canonical Basis Character Formula 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 [2308.16254] [1512.08296] [2603.01373]. ## 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'0\), the principal block \(\operatorname{Rep}_0(G)\) 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 \(p\)-canonical basis. The indicated formulas are \[ [T(\lambda)]\leftrightarrow {}^p\underline{N}_w \] for tilting modules and \[ [L(\lambda)]=\sum_\mu (\text{inverse }p\text{-KL coefficient})\,[\Delta(\mu)] \] schematically for simple modules. For \(G=\mathrm{GL}_n\), the conjectural action is proved via \(2\)-Kac–Moody actions, so the \(p\)-canonical basis character formulas hold in that case [1512.08296]. The comparison between canonical bases and decomposition numbers is especially explicit for Fock spaces and Schur algebras. Under restriction from \(U_q(\widehat{\mathfrak{gl}_n})\) to parabolic subalgebras \[ U(n)\cong U_q(\mathfrak{sl}_{n_1})\otimes\cdots\otimes U_q(\mathfrak{sl}_{n_r}), \] the Fock space decomposes as \[ F_s=\bigoplus_{t\in T_{n,s}} F_t, \] and each \(F_t\) is a based module carrying its own canonical basis. The comparison theorem states \[ T_t(G_s(\lambda))=G_t(\lambda), \qquad d_{\lambda\mu}(q)=d_{\lambda\mu}^t(q)\quad(\lambda,\mu\in\mathcal P_t), \] so the same canonical basis coefficients govern both the ambient and restricted modules. In the cases \(n_1,\dots,n_r\in\{1,2\}\) and \(\sum n_j=p\), these coefficients specialize at \(q=1\) to decomposition numbers of Schur algebras: \[ [P(\mu)]_t=\left.G_t(\mu)\right|_{q=1}, \qquad d_{\lambda\mu}=d_{\lambda\mu}^t(1). \] Runner-removal factorization gives product formulas for the coefficients [1503.03616]. The spin representation theory of symmetric groups furnishes a further example. In the level-\(1\) \(q\)-deformed Fock space of type \(A_{2n}^{(2)}\), canonical basis vectors \[ G\mu=\sum_\lambda d_{\lambda\mu}\,\lambda \] have coefficients \(d_{\lambda\mu}\in \mathbb Z[q]\), the “\(q\)-decomposition numbers.” For bar-weight \(1\) and \(2\) blocks, a Richards-style combinatorial formula determines these coefficients explicitly. In weight \(1\), \[ d_{\tau(r)\,\tau(s)}= \begin{cases} 1,& r=s,\\ q,& r=s+1\text{ and }h\in\tau(r),\\ q^2,& r=s+1\text{ and }h\notin\tau(r),\\ 0,& \text{otherwise,} \end{cases} \] while in weight \(2\) the formulas involve the \(\delta\)-statistic, colors, dominance intervals, and exceptional partitions \(Q\tau,K\tau,B\tau,N\tau,R\tau,p\tau\). The specialization at \(q=1\) is expected to recover reduced spin decomposition numbers in the corresponding defect-\(2\) blocks [1905.04080]. These examples show that the canonical basis character formula is often a decomposition-number formula in disguise: the basis coefficients, sometimes after evaluation at \(q=1\), encode standard-filtration multiplicities, simple characters, or block decomposition matrices. ## 4. Super and \(W\)-superalgebra formulations For Lie superalgebras and finite \(W\)-superalgebras, canonical basis character formulas appear as direct analogues of Kazhdan–Lusztig theory, but with tensor product models adapted to super phenomena. The finite \(W\)-superalgebra of type \(A\), denoted \(U(\mathfrak{gl},e)\), admits parabolic BGG-type categories \(P(\lambda)\) whose Grothendieck groups categorify tensor product modules of irreducible polynomial representations and their duals over \(U_q(\mathfrak{gl}_\infty)\). Standard modules \(\Delta(A)\) correspond to standard basis vectors \(\Delta_A\), irreducibles \(L(A)\) correspond to Lusztig’s dual canonical basis elements \(L_A\), and the basis-identification \[ \Psi([\Delta(A)])=\Delta_A,\qquad \Psi([L(A)])=L_A \] gives the character formula. The finite-dimensional category \(F(|\mu)\) is a special case: \[ \Psi:[F(|\mu)]^\wedge\to P^\lambda(+)\widehat\otimes P^\mu(-),\qquad \Psi([L(A^{(1)},A^{(2)})])=L_{(A^{(1)},A^{(2)})}. \] The theorem is presented as a uniform generalization of several earlier character formulas in BGG categories for Lie superalgebras and \(W\)-algebras of type \(A\) [2603.01373]. For \(\mathfrak q(n)\), two quantum-group types occur. In Brundan’s original formulation for integer weights, the conjectural character formula is expressed in terms of a type \(B\) canonical basis on tensor space: \[ [T(\lambda)]=\sum_\mu t_{\mu\lambda}(1)\,[A(\mu)],\qquad [L(\lambda)]=\sum_\mu \ell_{\mu\lambda}(1)\,[A(\mu)]. \] For half-integer weights, the relevant structure changes to type \(C\). The tensor space \(T^n=V^{\otimes n}\) and the type \(C\) \(q\)-wedge space \(F^n\) carry bar involutions defined by the quasi-\(R\)-matrix, with canonical and dual canonical bases \[ T_\lambda=M_\lambda+\sum_{\mu\prec \lambda} t_{\mu\lambda}(q)\,M_\mu,\qquad L_\lambda=M_\lambda+\sum_{\mu\prec \lambda} \ell_{\mu\lambda}(q)\,M_\mu. \] The finite-dimensional half-integer character formula is then \[ \Upsilon([L(\lambda)])=L_\lambda(1), \] which identifies the irreducible character with the specialized dual canonical basis element [1512.00116]. A different, but still canonical-basis-type, superalgebra phenomenon appears in the categorification of the generic Su–Zhang character formula for \(\mathfrak{gl}(m|n)\). For a dominant integral \(\mathfrak g_{-1}\)-generic weight \(\lambda\), the simple character is \[ \operatorname{ch} L^{\mathfrak b}(\lambda)= \sum_{w\in W}(-1)^{\ell(w)} \frac{\operatorname{ch} M^{\mathfrak b}(w\cdot\lambda)} {\prod_{\beta\in \Gamma_\lambda}(1+e^{-\beta})}. \] The categorifying object is the narrow Verma module \[ N^{\mathfrak b}(\lambda)=\operatorname{Im}\!\Bigl( M^{\mathfrak b}(\lambda)\to M^{(n^m)}(\lambda-2\rho_{\bar1}) \Bigr), \] whose character is the modified Verma character \[ \operatorname{ch} N^{\mathfrak b}(\lambda)= \frac{\operatorname{ch}M^{\mathfrak b}(\lambda)} {\prod_{\beta\in\Gamma_\lambda}(1+e^{-\beta})}. \] 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 [2512.16095]. ## 5. Quantum symmetric pairs and \(\imath\)-canonical bases Quantum symmetric pairs replace the usual quantum group by a coideal subalgebra \(U^\imath\), and the corresponding basis theory is governed by a twisted bar involution. For a quantum symmetric pair \((U,U^\imath)\) of arbitrary finite type, the \(\imath\)-canonical basis on a based \(U\)-module \((M,B)\) is the unique basis \[ B^\imath=\{b^\imath\mid b\in B\} \] such that \[ \psi^\imath(b^\imath)=b^\imath,\qquad b^\imath=b+\sum_{b'<_\imath b} t_{b;b'}\,b',\qquad t_{b;b'}\in q^{-1}\mathbb Z[q^{-1}]. \] The bar involution is defined using the intertwiner \(\Upsilon\), \[ \psi^\imath=\Upsilon\circ\psi, \qquad \psi^\imath(u)\,\Upsilon=\Upsilon\,\psi(u)\quad(\forall u\in U^\imath). \] Finite-dimensional simple \(U\)-modules, their tensor products, and the modified form \(\dot U^\imath\) all admit such \(\imath\)-canonical bases [1610.09271]. The triangular expansion \[ b^\imath=b+\sum_{b'<_\imath b} t_{b;b'}\,b' \] is the \(U^\imath\)-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, \[ (b_1^\imath,b_2^\imath)\equiv \delta_{b_1,b_2}\pmod{q^{-1}A}, \] which plays the same structural role as orthonormality in Lusztig’s characterization [1610.09271]. In rank one, the theory becomes completely explicit. For Letzter’s coideal subalgebra \(\mathbf U^\imath(\mathfrak{sl}_2)\cong \mathbb Q(v)[t]\) with \[ t=\mathbf E+v\,\mathbf K\mathbf F+\mathbf K, \] the modified form splits into even and odd pieces, each isomorphic to \(\mathbb Q(v)[t]\). The \(\imath\)-canonical basis elements have closed shifted-product formulas: \[ b_{0,d}= \frac{(t+[d-1])(t+[d-3])\cdots(t+[-d+1])}{[d]!}, \] \[ b_{1,d+1}= \frac{t\,(t+[d-1])(t+[d-3])\cdots(t+[-d+1])}{[d+1]!}. \] 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 \(t\) [1904.13340]. 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 [1610.09271] [1904.13340]. ## 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 \(G\), the canonical section \[ a_G:R(G)\to R_+(G) \] of the induction map yields a distinguished expansion \[ \chi=\sum_{[H,\varphi]_G}\alpha_{[H,\varphi]_G}(\chi)\,\varphi^G \] for any character \(\chi\). The derived invariant \[ S(G,\chi,n):=\sum_{n\mid o(\varphi)} \alpha_{[H,\varphi]_G}(\chi) \] is shown to equal a multiplicity in a virtual character built from Adams operations, \[ S(G,\chi,n)=\sum_{\rho\subseteq\pi(n)}(-1)^{|\rho|}\,(\Psi^{n(\rho)}(\chi),1_G), \] 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 [2510.03179]. Another analogue occurs in the trivial source algebra. The primitive idempotents \(e_{Q,s}\) of \(KTF(G)\) are expressed as linear combinations of the canonical \(\mathbb Z\)-basis \([N_{P,\varphi}]\): \[ e_{Q,s} = \frac{|N_G(Q)|}{|G|} \sum \frac{\varphi(gP)\,\mu_g(P,Q)}{|N_G(P)|}\,[N_{P,\varphi}], \] 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 [1809.10984]. Refined Weyl character formulas provide a different boundary case. For \(\operatorname{GL}_{2,A}\), the symmetric power \(\operatorname{Sym}^d(V)\) admits a canonical refined weight-space decomposition \[ \operatorname{Sym}^d(V)\cong \bigoplus_i \operatorname{Sym}^d(V)^i \] as comodules for the schematic normalizer \(N\) of the diagonal torus. The refined character formula is \[ \operatorname{Char}(\operatorname{Sym}^d(V)^i)=x^{d-i}y^i+x^iy^{d-i} \] for paired Weyl-orbit pieces, with the even-degree middle term \(x^ky^k\) in degree \(2k\). 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 [2405.09210]. 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 \[ [e_\alpha^\varepsilon,e_\beta^\varepsilon] = \eta_\varepsilon(\alpha,\beta)\,(q_{\alpha,\beta}+1)\,e_{\alpha+\beta}^\varepsilon. \] 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 [2404.07652]. 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. Source: https://www.emergentmind.com/topics/canonical-basis-character-formula