Orbifold Type Decomposition Theorem

• The orbifold type decomposition theorem decomposes Hochschild homology of symmetric quotient stacks into direct sums indexed by symmetric group cycle types.
• It employs explicit chain-level functorial methods and averaging techniques in dg categories to bridge equivariant geometry with noncommutative invariants.
• The theorem underpins induced algebraic structures such as Fock space representations, Hopf algebras, and free λ-rings, with applications to intersection cohomology.

The orbifold type decomposition theorem provides a structure theorem for the invariants—most notably Hochschild homology—of symmetric quotient stacks, both in the classical algebro-geometric and in the noncommutative setting. A modern formulation encompasses the decomposition of homological invariants of the symmetric powers of differential graded (dg) categories, identifying a direct sum indexed by cycle types of the symmetric group, with explicit functorial isomorphisms to symmetric powers of the invariants of the base category. These results generalize classical orbifold cohomology calculations of quotient varieties and elucidate the interplay between equivariant geometry, noncommutative motives, and representation theory.

1. Noncommutative Symmetric Quotient Stacks and Derived Categories

For a small dg $k$-category $\mathcal{A}$ (interpreted as a noncommutative "space") over a field $k$ of characteristic zero, the $n$th symmetric quotient stack is defined as

$\Sym^n\mathcal{A} := \mathcal{A}^{\otimes n} \rtimes S_n,$

where $S_n$ acts by permutation on the $n$-fold tensor product. The objects of $\Sym^n\mathcal{A}$ are $n$-tuples of objects in $\mathcal{A}$, morphisms are twisted by summing over group elements, and the structure realizes the categorical analog of the quotient stack $[X^n/S_n]$ for schemes $X$.

Morita-localization identifies $\Sym^n\mathcal{A}$ with the $S_n$-equivariant category $\Perf(\mathcal{A}^{\otimes n})^{S_n}$. This formulation allows the extension of classical symmetric power constructions to dg categories, integral to noncommutative geometry and applications to intersection cohomology via resolution categories associated to stacks.

2. Orbifold-Type Decomposition Theorem: Statement and Construction

Given a symmetric quotient stack as above, the orbifold type decomposition theorem for Hochschild homology asserts a canonical isomorphism

$\HH_*(\Sym^n\mathcal{A}) \cong \bigoplus_{\lambda \vdash n} \Sym^{r(\lambda)} \HH_*(\mathcal{A})$

where $\lambda$ runs over partitions of $n$, $r(\lambda)$ denotes the multiplicities of cycle lengths, and $\Sym^m$ denotes the $m$th symmetric power. The key steps in establishing this decomposition are as follows:

• For a finite group $G$ acting strongly on a dg category $\mathcal{C}$, the Hochschild homology of the crossed product $\mathcal{C}\rtimes G$ admits a chain-homotopy equivalence to the sum over twisted sectors (homology with coefficients twisted by group elements), modulo $G$-conjugation:

$\HH_*(\mathcal{C}\rtimes G) \simeq \left(\bigoplus_{g\in G} \HH_*(\mathcal{C};g)\right)_G.$

• For $G = S_n$ and $\mathcal{C} = \mathcal{A}^{\otimes n}$, one can further refine the sum over conjugacy classes (cycle types), factorizing each sector by Künneth formula to ordinary Hochschild homology of $\mathcal{A}$.
• Explicit chain-level equivalences are built using averaging over cyclic permutations and insertion of identity morphisms, making the entire decomposition functorial and constructing mutually inverse homotopy equivalences (Anno et al., 31 Dec 2025).

The upshot is that the invariants of the symmetric quotient stack are completely determined by those of the base category, organized according to the classical combinatorics of $S_n$.

3. Symmetric Algebra Structure and Total Hochschild Homology

Summing over all $n \ge 0$, the decomposition further gives rise to a canonical graded isomorphism

$\bigoplus_{n\ge0}\HH_*(\Sym^n\mathcal{A}) \xrightarrow{\;\simeq\;} S^*(\HH_*(\mathcal{A}) \otimes t\,k[t]),$

where $k[t]$ is the polynomial algebra in a formal variable $t$ and $S^*$ is the free symmetric algebra functor. The variable $t$ encodes the partition data, with $\HH_*(\mathcal{A}) \otimes t^j$ corresponding to cycles of length $j$. This structure enables the direct calculation of the entire tower of symmetric quotients in terms of the base homological data (Anno et al., 31 Dec 2025).

The table below summarizes the key correspondences for individual $n$:

$n$	Decomposition of $\HH_*(\Sym^n\mathcal{A})$	Algebraic Structure
$n=1$	$\HH_*(\mathcal{A})$	Identity
$n=2$	$\Sym^2\HH_*(\mathcal{A}) \oplus \HH_*(\mathcal{A})$	Partition: $(2), (1,1)$
all $n$	$\bigoplus_{\lambda\vdash n} \Sym^{r(\lambda)}\HH_*(\mathcal{A})$	$S^*(\HH_*(\mathcal{A}) \otimes t\,k[t])$

This explicit description of total Hochschild homology admits natural algebraic structures transported from the symmetric algebra, reflecting the deeper symmetries of the orbifold setting.

4. Induced Algebraic and Representation-Theoretic Structures

Three fundamental algebraic structures are induced on the total Hochschild homology via the explicit isomorphism with the symmetric algebra:

• Fock Space/Heisenberg Algebra: Creation and annihilation operators arising from adding or removing tensor factors induce a Fock space representation of the Heisenberg algebra of $\HH_*(\mathcal{A})$. The commutation relation

$$[\alpha^+_n, \alpha^-_m] = n\,\delta_{n,m}\,\mathrm{id}$$

holds, generalizing the classical structure of symmetric functions and Nakajima's picture of Hilbert schemes.

• Hopf Algebra: The product is given by induced shuffles and group induction, the coproduct by restriction and Alexander–Whitney decomposition; the symmetric algebra's standard Hopf structure is thus realized on the noncommutative Hochschild invariants.
• Free $\lambda$-Ring Structure: Adams operations act via correspondences indexed by cycle type, with the action on generators matching their degree; specifically, $\psi^m(x t^j) = x t^{mj}$ for $x \in \HH_*(\mathcal{A})$.

These structures generalize classical constructions for varieties and stacks to the setting of dg categories, providing a robust algebraic toolkit for further developments in equivariant and homological invariants (Anno et al., 31 Dec 2025).

5. Relation to Noncommutative Resolutions and Intersection Cohomology

In the algebro-geometric context, orbifold type decompositions inform the structure of noncommutative resolutions of quotient singularities. For a symmetric stack $\mathcal{X}$ with good moduli space $X$, Pădurariu constructs a global noncommutative resolution $D(X) \subset D^b(\mathcal{X})$ with adjoint functors to $D^b(X)$, providing a categorification of intersection cohomology: $$HP_*(\mathbb{D}^{nc}(X)) \cong IH^*(X, \mathbb{C}),$$ where $\mathbb{D}^{nc}(X)$ is a direct summand of $D(X)$ in the category of noncommutative motives, split by an explicit idempotent. This construction recovers the "window" subcategories of Špenko–Van den Bergh and globalizes them to a large class of quotient stacks (Pădurariu, 2021).

The orbifold decomposition theorem for Hochschild homology operates in parallel to these geometric resolutions, providing the homological and representation-theoretic infrastructure underlying the realization of intersection cohomology and related invariants in terms of symmetric and twisted sectors.

6. Explicit Functoriality and Chain-Level Description

The chain-level equivalences underpinning the orbifold type decomposition are constructed by explicit averaging and insertion maps, as well as classical Eilenberg–Zilber and Alexander–Whitney maps for shuffles and (co)products. These constructions ensure functoriality across dg categories and compatibility with group actions. Homotopy inverses and degree $-1$ homotopies such as $\Phi_n$ and $\Psi_n$ provide explicit chain homotopies verifying that the relevant functors split idempotently and recover the symmetric algebra structure up to homotopy (Anno et al., 31 Dec 2025).

A plausible implication is that this explicit control over the decomposition can be extended to multiplicative filtrations and further categorical refinements, deepening the understanding of higher structures in both commutative and noncommutative equivariant contexts.

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The orbifold type decomposition theorem thus synthesizes combinatorial representation theory, noncommutative geometry, and derived algebraic geometry to provide a comprehensive structural theorem for the invariants of symmetric quotient stacks, with significant implications for K-theory, cyclic homology, and categorifications of intersection cohomology. Its explicit, functorial form facilitates further generalizations and applications in modern homological approaches to algebraic and noncommutative geometry (Pădurariu, 2021, Anno et al., 31 Dec 2025).