Schur's Double Cover in Representation Theory

• Schur's double cover is a central extension of the symmetric group Sₙ by a cyclic group of order 2, enabling spin and projective representations.
• It underpins the classification of modular blocks, with RoCK blocks exhibiting graded Morita and derived equivalences to generalized Schur superalgebras.
• The structure also manifests in combinatorial generating functions and double series, linking algebraic and analytic approaches in partition theory.

Schur’s double cover is a foundational construction in algebraic combinatorics, representation theory, and partition theory, manifesting in multiple forms across group extensions, spin representation theory, and the combinatorics of partitions. At its core, Schur’s double cover appears as a central extension of the symmetric (or alternating) group by a cyclic group of order two, giving rise to spin or projective representations, but the influence of this doubling is also found in the analytic and combinatorial structure of generating functions and beyond.

1. Central Extension and Spin Representations

Schur’s double cover of the symmetric group $S_n$, typically denoted $\widetilde{S}_n$, is defined via the exact sequence

$$1 \to \langle z \rangle \to \widetilde{S}_n \to S_n \to 1,$$

where $z$ is a central element of order 2. This extension arises when classifying projective representations of $S_n$: every projective representation corresponds to a genuine linear representation of $\widetilde{S}_n$ (the double cover), called a spin representation. The defining relations for these double covers involve generating elements subject to sign-twisted commutation and braid-type relations, such as

$$t_r^2 = 1, \quad t_r t_s = - t_s t_r \ (|r-s| > 1), \quad (t_r t_{r+1})^3 = 1,$$

where the $t_i$ are odd generators for the twisted group algebra (see Example 3.25), encapsulating the additional "spin" structure not present in $S_n$.

Spin representations induced by the double cover are critical for understanding the entirety of projective modular representation theory of symmetric and alternating groups, as all irreducible projective representations factor through such a cover.

2. RoCK Blocks and Generalized Schur Superalgebras

Within the modular representation theory of $\widetilde{S}_n$, the structure of its group algebra decomposes into blocks. Of specific interest are the RoCK blocks, parametrized by a $p$-core partition $\rho$ and weight $d$, and characterized by pairing conditions such as

$$\langle \theta, \alpha_0 \rangle \ge 2d, \quad \langle \theta, \alpha_i \rangle \ge d-1, \quad \theta = \text{cont}(\rho) + d\delta,$$

with $h(\rho)$ denoting the number of nonzero parts in the core partition, and $\alpha_i$ simple roots.

A principal result is that every such RoCK block $B_{\rho, d}$ of the double cover is graded Morita superequivalent (or derived equivalent) to a generalized Schur superalgebra $T^{A_\ell}(d, d)$, with a possible tensor factor of a rank-one Clifford superalgebra when the parity $|\rho| - h(\rho) + d$ is odd: $$B_{\rho, d} \sim_{sMor} \begin{cases} T^{A_\ell}(d, d) & \text{if } |\rho| - h(\rho) + d \text{ even} \ T^{A_\ell}(d, d) \otimes C & \text{if } |\rho| - h(\rho) + d \text{ odd} \end{cases}$$ where $C$ is a suitable Clifford algebra. Generalized Schur superalgebras realize a super-analogue of Schur–Weyl duality, capturing the $\mathbb{Z}/2$-grading imposed by the double cover structure and enabling the explicit description of block representation theory.

3. Double Series and Combinatorial Double Cover Phenomena

In combinatorics, Schur's double cover manifests within the structure of generating functions counted by partition identities. Notably, the double series representation of Schur’s partition generating function $f_S(x;q)$ inherently "doubles" the variables of summation: $$f_S(x;q) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} (-1)^n x^{m+2n} \frac{q^{3n(3n+2m)+\cdots}}{(q;q)_m (q^6;q^6)_n}.$$ This decomposition mirrors the double cover phenomenon: one summation index corresponds to the "core" components after removing a triangular block (in the 3-modular diagram), and the other tracks the contributions of that triangle. The expansion into a double sum encodes the layered combinatorial structure that reflects the twofold character of the double cover at the level of enumeration.

Methodologically, bijective mappings and modular diagrams afford a transparent route to this decomposition, while $q$-difference equations analytically confirm the double series' validity. The interplay between combinatorial "lifting" and analytic recurrence thus reenacts the double cover as an organizing principle within partition theory (Andrews et al., 2014).

4. Morita and Derived Equivalences: Local Models and Broué’s Conjecture

The derived equivalence results between RoCK blocks of double covers and generalized Schur superalgebras are significant for local representation theory and Broué’s Abelian Defect Group Conjecture. The latter predicts that blocks with abelian defect groups are derived equivalent to their Brauer correspondent in the normalizer. The explicit description,

$B_{\rho, d} \sim_{sMor} T^{A_\ell}(d, d)$

(possibly with a Clifford twist), establishes that every spin block is not just "covered" by a generalized Schur superalgebra but that this cover is computable and combinatorial in nature. This "localization" of block theory directly supports the realization of Broué’s conjecture in the field of spin representations for the double cover, providing an explicit and tractable endomorphism algebra model for each block (Kleshchev, 6 Nov 2024).

5. Double Cover, Essential Dimension, and Representation Complexity

The structure of Schur's double cover also heavily impacts the essential dimension of groups and their representations. For the double covers $\widetilde{S}_n^{\pm}$ and their alternating group analogues, the essential dimension at the prime 2 is given by

$$\operatorname{ed}(\widetilde{S}_n^{\pm};2) = 2^{\lfloor (n-s)/2 \rfloor}, \quad \operatorname{ed}(\widetilde{A}_n;2) = 2^{\lfloor (n-s-1)/2 \rfloor},$$

where $s$ is the number of nonzero terms in the dyadic expansion of $n$. This exponential growth (in characteristic not 2) reflects the complexity induced by the spin structure, while in characteristic 2, the invariant collapses to a sublinear regime, demonstrating the sensitivity of essential dimension to the nature of the covering (Reichstein et al., 2019).

These results also have ramifications in quadratic form theory — for example, the maximal splitting index of the Hasse invariant for trace forms of étale algebras matches the essential dimension at 2 of the double cover, linking group-theoretic and arithmetic invariants.

6. Extensions and Related Double Cover Structures

The double cover principle appears analogously in various algebraic structures:

• Turner doubles: In (Evseev et al., 2016), Turner doubles are maximal symmetric subalgebras of certain generalized Schur algebras. Their construction doubles both an invariant algebra and its dual (via a coproduct structure), providing a category-theoretic model generalizing double cover principles.
• Partial projective representations and partial Schur multipliers: Schur’s double cover concept is generalized to partial actions, leading to the notion of a partial Schur multiplier, governed by a semilattice of partial cohomology groups. The resulting central extensions and cohomological obstructions directly analogize classical double covers, but in the context of partial representations (Dokuchaev et al., 2017).
• Branched invariants and topological covers: In (Samperton, 2017), Schur-type invariants for branched $G$-covers of surfaces depend on the Schur multiplier, giving rise to obstructions and invariants which generalize the double cover phenomenon in algebraic topology and topological quantum field theory.

7. Broader Significance and Ongoing Research

The impact of Schur’s double cover is profound, intertwining algebraic, combinatorial, and topological perspectives. The passage from projective to spin representations, the manifestation of averaging over double cosets or summing over combinatorial decompositions, and the translation into superalgebraic frameworks are all structurally governed by the double cover philosophy. The explicit realization of derived and Morita equivalences for RoCK blocks provides not only evidence for significant conjectures but also computational and categorical tools for further exploration. The continued expansion into new algebraic structures (e.g., relative Rota–Baxter groups and their Schur covers (Belwal et al., 2023)) suggests that the “double cover” framework will persist as a unifying concept for organizing and elucidating otherwise complex representation-theoretic and cohomological phenomena.

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Table: Key Equivalences for RoCK Blocks of Double Covers

Block Type	Local Model	Clifford Twist Condition
$B_{\rho,d}$	$T^{A_\ell}(d,d)$	Absent if $|\rho| - h(\rho) + d$ even; present if odd
Spin Block	Derived equiv. to generalized Schur superalgebra	Parity on core data

These correspondences—proven up to graded Morita superequivalence and derived equivalence—encode the "localization" of block theory in the setting of Schur's double cover (Kleshchev, 6 Nov 2024).