Representation theory of projective Clifford groups via isocategoricality

Abstract: The representation theory of the projective Clifford group $C(A)$, attached to a finite abelian group $A$, is closely related to the symplectic action on $V_A=A\oplus\widehat A$. We make this relation precise by constructing an explicit tensor isomorphism between the representation category of $C(A)$ and the representation category of the affine symplectic group $$\operatorname{ASp}(A)=\operatorname{Sp}(V_A)\ltimes\widehat{V_A}$$. Thus $C(A)$ and $\operatorname{ASp}(A)$ are isocategorical, although they need not be isomorphic. The isomorphism transfers the little-group method from $\operatorname{ASp}(A)$ to $C(A)$, giving a uniform description of the irreducible representations of $C(A)$. The same approach gives conjugacy-class parameters, class-size formulas, and character formulas. Thus the character theory of $C(A)$ is reduced to ordinary character tables of stabilizers, affine centralizer orbits, and the scalar factors appearing in the Clifford action. In particular, $C(A)$ and $\operatorname{ASp}(A)$ have identical ordinary character tables, up to relabeling. Finally, the tensor isomorphism identifies the twisted group algebra determined by the Weyl commutation relations with the tensor transport of the ordinary group algebra $\mathbb C[V_A]$. It also transports the Clifford adjoint-action commutants to affine symplectic orbit algebras, where they admit an orbit basis with orbit-intersection structure constants.

• The paper introduces a tensor equivalence between Rep(C(A)) and Rep(ASp(A)) that shifts complex computations in non-split projective cases to the more tractable affine symplectic framework.
• It parametrizes irreducible representations using Sp(V_A) orbits and applies the little-group method, thereby confirming the Basheer–Moori conjecture for elementary abelian 2-groups.
• Explicit formulas for conjugacy classes and character values are derived by transporting algebraic structures with a Heisenberg bicharacter twist, impacting quantum information and coding theory.

Representation Theory of Projective Clifford Groups via Isocategoricality

Overview

This paper develops a comprehensive framework for the representation theory of projective Clifford groups $C(A)$ associated to finite abelian groups $A$, establishing a canonical tensor equivalence between $\mathrm{Rep}(C(A))$ and the representation category of the affine symplectic group $$\mathrm{ASp}(A)=\mathrm{Sp}(V_A)\ltimes\widehat{V_A}$$, with $V_A = A \oplus \widehat{A}$. The construction leverages isocategoricality, providing explicit correspondences for irreducibles, conjugacy classes, and character formulas, as well as a mechanism to transfer structure and computations from the more tractable affine case to the possibly non-split projective Clifford groups.

Main Results and Methods

Tensor Equivalence and Isocategoricality

The paper's principal contribution is the explicit construction of a tensor equivalence $$F: \mathrm{Rep}(\mathrm{ASp}(A)) \to \mathrm{Rep}(C(A))$$. This equivalence acts as the identity on underlying vector spaces equipped with a $V_A$-grading and $\mathrm{Sp}(V_A)$-action, but twists the tensor product by a Heisenberg bicharacter $\beta_A$, determined by the Weil commutation relations. The tensor structure is explicitly described: the underlying vector space is preserved, while the action of $C(A)$ is transported via $A$0 for $A$1 homogeneous.

The equivalence respects the symmetric monoidal structures when the standard symmetry in $A$2 is replaced by a symmetry twisted by the canonical symplectic form $A$3, reflecting the physical significance of commutation relations in Clifford theory.

Crucially, this isocategoricality holds even when the central extension

$A$4

is non-split, notably when $A$5 divides $A$6. In this setting, $A$7 and $A$8 are not isomorphic as groups, but their representation categories are tensor equivalent.

Uniform Classification of Irreducibles

Applying this equivalence, irreducible representations of $A$9 are parametrized in direct analogy with the affine symplectic case: they are associated to pairs $\mathrm{Rep}(C(A))$0 with $\mathrm{Rep}(C(A))$1 a representative of a $\mathrm{Rep}(C(A))$2-orbit in $\mathrm{Rep}(C(A))$3 and $\mathrm{Rep}(C(A))$4 an irreducible of the stabilizer. The paper demonstrates, using the little-group method, that every irreducible of $\mathrm{Rep}(C(A))$5 arises as an induction from a linear extension of a character of the kernel, twisted by the parametrizing data from the affine side.

The induced realization and explicit extension of the kernel character are constructed, confirming that the resulting representations are governed by ordinary (not projective) character tables of the relevant stabilizers. This result is applied to settle the Basheer--Moori conjecture for elementary abelian $\mathrm{Rep}(C(A))$6-groups, confirming that projective character tables are unnecessary for specific nontrivial blocks in the Clifford group’s character table.

Conjugacy Classes and Character Formulas

A precise parameterization of the conjugacy classes of $\mathrm{Rep}(C(A))$7 is developed. Above each $\mathrm{Rep}(C(A))$8 (modulo conjugacy), classes are labeled by affine orbits in $\mathrm{Rep}(C(A))$9 under a centralizer action twisted by a cocycle derived from the splitting section. The class size and centralizer formulas are given in terms of this data.

The paper derives explicit formulas for irreducible characters on class representatives. For each $$\mathrm{ASp}(A)=\mathrm{Sp}(V_A)\ltimes\widehat{V_A}$$0 and class representative $$\mathrm{ASp}(A)=\mathrm{Sp}(V_A)\ltimes\widehat{V_A}$$1, the value is

$$\mathrm{ASp}(A)=\mathrm{Sp}(V_A)\ltimes\widehat{V_A}$$2

which depends on the symplectic form, section factors, and stabilizer characters, but crucially does not require projective tables.

Transport of Algebraic Structures

The paper identifies the twisted group algebra $$\mathrm{ASp}(A)=\mathrm{Sp}(V_A)\ltimes\widehat{V_A}$$3 realized by Weyl operators with the transport of the group algebra $$\mathrm{ASp}(A)=\mathrm{Sp}(V_A)\ltimes\widehat{V_A}$$4 under $$\mathrm{ASp}(A)=\mathrm{Sp}(V_A)\ltimes\widehat{V_A}$$5. Tensor constraints insert the factor $$\mathrm{ASp}(A)=\mathrm{Sp}(V_A)\ltimes\widehat{V_A}$$6 in multiplication, altering commutativity in the Morita context to encode the Weyl commutation.

Tensor powers and their commutant algebras are also transported: the commutant of the adjoint $$\mathrm{ASp}(A)=\mathrm{Sp}(V_A)\ltimes\widehat{V_A}$$7-action on $$\mathrm{ASp}(A)=\mathrm{Sp}(V_A)\ltimes\widehat{V_A}$$8 is described as an orbit algebra indexed by $$\mathrm{ASp}(A)=\mathrm{Sp}(V_A)\ltimes\widehat{V_A}$$9-orbits in $V_A = A \oplus \widehat{A}$0, with structure constants expressed via explicit orbit-intersection numbers.

Families and Applications

Cyclic and Elementary Abelian Cases

The framework is elucidated for important families: when $V_A = A \oplus \widehat{A}$1, the structure is governed by the $V_A = A \oplus \widehat{A}$2-adic valuation filtration; for $V_A = A \oplus \widehat{A}$3, $V_A = A \oplus \widehat{A}$4 acts doubly transitively, resulting in only two $V_A = A \oplus \widehat{A}$5-orbits in $V_A = A \oplus \widehat{A}$6. In these cases, block structures, stabilizers, and explicit character values are detailed, demonstrating the generality and utility of the isocategorical approach.

The confirmation of the Basheer--Moori conjecture has further implications for computational group theory and for the study of character tables of groups arising in quantum information and coding theory.

Theoretical and Practical Implications

The construction situates the projective Clifford groups $V_A = A \oplus \widehat{A}$7 within the paradigm of isocategorical but non-isomorphic pairs, augmenting the understanding of the relationships between quantum symmetries, Heisenberg groups, and symplectic geometry. By reducing computations in the possibly non-split, projectively defined $V_A = A \oplus \widehat{A}$8 to the affine semidirect product $V_A = A \oplus \widehat{A}$9, the results allow for more tractable analysis of representation theory, character tables, and commutant algebras, which are central to numerous quantum information processing tasks.

In particular, the tensor equivalence provides a transport mechanism not only for representation-theoretic and character-theoretic data but also for underlying algebraic and combinatorial structures, such as commutant algebras and orbit algebras found in higher tensor powers, relevant for understanding Schur--Weyl dualities, unitary designs, and stabilizer measurements.

Future directions include extension of this framework to more general groups, applications to quantum error correction and randomized benchmarking, and further exploration of isocategorical phenomena in non-semisimple and infinite settings.

Conclusion

This paper establishes that the representation category of $$F: \mathrm{Rep}(\mathrm{ASp}(A)) \to \mathrm{Rep}(C(A))$$0 is tensor equivalent to that of the corresponding affine symplectic group, regardless of the (non)splitting of the Clifford extension, with a fully explicit functorial correspondence. This unifies and extends the understanding of Clifford groups, their representations, and character theory, and resolves longstanding computational and theoretical questions regarding blocks and inertia factors, exemplified by the Basheer--Moori conjecture. The results have deep implications for the structure, computation, and application of Clifford group symmetries in mathematics and quantum theory.