Split Grothendieck Group Invariants

• The split Grothendieck group is a refined invariant defined by considering only split exact sequences, uniquely capturing direct sum decompositions in additive categories.
• It bridges local and global representation theory by yielding canonical decompositions that reflect subtle structures like centralizers, blocks, and combinatorial parameters.
• The invariant finds applications in silting, cluster tilting, and categorification frameworks, providing a unified tool for classifying subcategories and calculating invariants.

The split Grothendieck group is a refined invariant in additive, triangulated, and extriangulated categories, capturing direct sum decomposition behavior via generators and relations that reflect only split exact sequences. In many advanced categorical contexts—including Frobenius categories, silting and cluster tilting subcategories, distribution algebras, and representation theory—the split Grothendieck group exhibits canonical decompositions corresponding to finer structures such as centralizers, blocks, or combinatorial parameters. Its key role is in encoding and determining the algebraic structure of subcategories, the classification of categorical ideals, and—in suitable settings—bridging local and global representation-theoretic phenomena.

1. Definition and Construction

The split Grothendieck group, typically denoted $K_0^{\mathrm{sp}}(\mathcal{A})$, is defined for an additive category $\mathcal{A}$ by taking the free abelian group generated by isomorphism classes $[A]$ of the objects $A$ in $\mathcal{A}$, subject only to the relation $[A \oplus B] = [A] + [B]$. Unlike the classical Grothendieck group, which imposes further relations arising from distinguished triangles (in triangulated categories) or short exact sequences (in exact categories), the split version only encodes direct sum decomposability. The resulting group is typically free abelian, with one generator for each indecomposable iso-class.

If $\mathcal{A}$ admits further structure (such as a monoidal or triangulated structure), variants of the split group may include more refined relations or be equipped with ring structures (as in tensor categories), but the essential characteristic remains that only split decompositions, rather than general extensions, are considered.

2. Splitting via Centralizer Decomposition in Frobenius Categories

In the context of Frobenius $P$-categories over a finite $p$-group $P$, the ordinary Grothendieck group is constructed as an inverse limit over localization data (e.g., chains of self-centralizing subgroups). After extension of scalars to a field, the global Grothendieck group canonically splits as a direct sum over contributions from centralizer subcategories: $$KGK(F,\operatorname{aut}^{sc}_F) \cong \bigoplus_{u\in\mathcal{P}} KGK(C_F(u),\operatorname{aut}^{sc}_{C_F(u)}),$$ where $\mathcal{P}$ is a set of representatives for $F$-classes of elements in $P$ (Puig, 2010). This decomposition is canonical, functorial, and independent of choices. The rank of the group is determined by an alternating sum over regular chains, reminiscent of Euler characteristics in algebraic topology: $$\operatorname{rank}_O(GK(F,\operatorname{aut}^{sc}_F)) = \sum_{(q,A_n)}(-1)^n \operatorname{rank}_O(GK(\hat{F}(q))).$$ This splitting reflects the deep local-global relationship in modular representation theory and can be used to compute invariants such as the number of simple modules or block decompositions, paralleling the structure seen in representation rings.

3. Isomorphism of Split and Triangulated Grothendieck Groups

In additive categories, the split Grothendieck group $K_+(\mathcal{A})$ and the triangulated Grothendieck group $K_\triangle(K^b(\mathcal{A}))$ of the homotopy category are canonically isomorphic (Rose, 2011). The isomorphism arises because every class in the triangulated group is determined by the Euler characteristic (the alternating sum of the terms of the complex), and relations from distinguished triangles do not introduce new relations beyond those already present in the split group: $$\langle A^\bullet \rangle = \sum_i (-1)^i \langle A^i \rangle.$$ This implies that the process of passing from objects to complexes and taking homotopy equivalence collapses to the computation of split decomposition data via the Euler characteristic. The invariance of the Euler characteristic under homotopy equivalence underlies this identification, essential for decategorification in contexts such as knot theory and categorification frameworks.

4. Applications in Silting, Cluster-Tilting, and Extriangulated Categories

In silting theory, if $\mathcal{M}$ is a silting subcategory of a triangulated category $\mathscr{C}$, then

$$K_0(\mathscr{C}) \cong K_0^{\mathrm{sp}}(\mathcal{M}),$$

with an explicit isomorphism constructed via filtrations whose factors belong to shifts of $\mathcal{M}$ (Wang, 18 Aug 2025, Chen et al., 24 Apr 2024). In the cluster category framework, for a $d$-cluster tilting subcategory, $K_0(\mathscr{C})$ is isomorphic to a quotient or index Grothendieck group derived from split data in the subcategory, often necessitating correction terms built from exchange triangles. These isomorphisms create a bridge between the global K-theoretic invariants of the category and the combinatorial data in distinguished subcategories, streamlining the classification of substructures and support theories.

Special cases such as the cluster category of type $A_n$ yield explicit computations:

• If $d$ even: $$K_0(\mathcal{C}_{A_n^d}) \cong \mathbb{Z}/(n+1)\mathbb{Z}$$.
• If $d$ odd: $K_0(\mathcal{C}_{A_n^d}) \cong \mathbb{Z}$ for $n$ odd; $K_0(\mathcal{C}_{A_n^d}) \cong 0$ for $n$ even.

These patterns reflect parity and combinatorial properties that are tightly coupled to the underlying categorical structure.

5. Split Grothendieck Groups in Monoid Representations and Categorification

The split Grothendieck ring appears in the context of monoid representations over $\mathbb{F}_1$ ("field of one element"), where modules are pointed sets and indecomposable objects correspond to combinatorial structures such as rooted trees, wheels, and skew shapes (Beers et al., 2018). Multiplication in the ring arises from the smash product, and every module decomposes uniquely into indecomposables. The explicit combinatorial algebra arising from these constructions links the split Grothendieck group to spectral graph theory (via adjacency matrices and Jordan decomposition after base-change to fields) and provides a model for linear algebra over $\mathbb{F}_1$. These results contribute to the paper of Hall algebras, categorification, and cluster algebras.

The nil-Brauer category provides a diagrammatic categorification of the split $\imath$-quantum group of rank one, establishing an isomorphism between its Grothendieck ring and an integral form of the quantum group. Indecomposable projectives correspond to the $\imath$-canonical basis, and new PBW-type bases are categorified by standard modules (Brundan et al., 2023).

6. Split Decomposition and Canonical Bases in Representation Theory

Lusztig's results on Grothendieck groups of unipotent representations for split and nonsplit reductive groups over finite fields demonstrate highly structured bases, e.g., the "second basis," which is canonical, upper triangular, and exhibits favorable positivity and Fourier transform properties. These bases split the Grothendieck group into direct summands, each well-controlled by combinatorial data such as families of the Weyl group and symbols parametrizing representations (Lusztig, 2019, Lusztig, 2022). In the modular setting, presentations as quotients of polynomial rings (e.g., $\mathbb{Z}[x]/(f^{(g)}(x) - x)$ for $SL_2(\mathbb{F}_q)$) reveal that split Grothendieck groups are determined as products over irreducible factors, reflecting the full decomposition of representation categories (Reduzzi, 2011).

In stacks of G-zips, the Grothendieck group splits as a quotient of the representation ring of the Levi subgroup by an ideal generated by Frobenius difference relations (Cooper, 2 Oct 2024).

7. Classification, Subgroup Correspondence, and Homological Applications

A major function of the split Grothendieck group is its role in universal classification theorems: subgroups of the split group classify dense and complete subcategories (or tensor ideals) in n-angulated, n-exangulated, exact, or triangulated categories for odd n (Bergh et al., 2012, Haugland, 2019). In the tensor setting, ideals of the Grothendieck ring correspond to tensor ideals of the category; in higher homological algebra, split Grothendieck group data underpin classification of subcategories and enable detection of local substructures directly from algebraic K-theory.

For n-cluster tilting subcategories, split group relations arise from alternating sums over n-almost split sequences, and the basis of the kernel associated to the projection from split to full group is determined by the existence of finite type or additive generators (Diyanatnezhad et al., 2021). Such classification mechanisms extend and unify prior work in triangulated and exact categories and are fundamental for understanding support theories, representation type, and categorification programs.

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The split Grothendieck group is thus a central invariant in algebraic, homological, and representation-theoretic settings, providing the foundational tool for decomposition, classification, and structural understanding of categories—bridging local, combinatorial data and global algebraic invariants. Its properties, particularly in canonical splitting or classification correspondence, are confirmed and utilized in diverse categorical frameworks, including those with silting and cluster tilting substructures, monoidal and periodically-derived categories, and algebraic stacks.