Source Algebra Equivalence

Updated 11 January 2026

Source algebra equivalence is an equivalence relation defined by algebraic invariants, such as Brauer trees and endo-permutation modules, that classify cyclic p-blocks in finite group algebras.
In modular representation theory, it uniquely determines the structure of cyclic p-blocks by pairing planar Brauer trees with Linckelmann’s Dade invariants, leading to explicit classification results.
In functional program semantics, source algebra equivalence underlies verification processes, ensuring that original and transformed code remains behaviorally identical through algebraic rewriting rules.

A source algebra equivalence fundamentally refers to the equivalence relation between algebraic or programmatic objects governed by the structure of a "source algebra" and its invariants. This concept is central in both modular representation theory—where it encodes a fine class of block equivalence for finite group algebras—and in program semantics and verification, where it formalizes when two code fragments or rewriting systems are "the same" up to transformations allowed by certain source-level algebraic rules. Two areas in particular crystallize its meaning: (1) the classification of $p$ -blocks with cyclic defect groups in modular representation theory, and (2) the verification of functional program or source code equivalence under algebraic transformations.

1. Source Algebra Equivalence in Modular Representation Theory

Let $G$ be a finite group, $B$ a $p$ -block of the group algebra $kG$ with cyclic defect group $D$ , over an algebraically closed field $k$ of characteristic $p > 0$ . The source algebra $A(B)$ of $B$ is the $D$ -interior subalgebra $e(kG)e$ with $e$ a source idempotent in a maximal Brauer pair. Source algebra equivalence between blocks $B$ of $kG$ and $B'$ of $kH$ with isomorphic defect groups is defined as the existence of a $D$ -interior algebra isomorphism $A(B) \cong A(B')$ , after choosing compatible identifications of their defect groups. This equivalence is strictly finer than Morita equivalence, coinciding only in special cases (notably for cyclic defect blocks) and capturing additional module-theoretic structure relevant for block theory and fusion systems (Hiss et al., 2022, Hiss et al., 13 Feb 2025, Hiss et al., 4 Jan 2026).

2. Invariant Structure: The Brauer Tree and Linckelmann's Dade Invariant

For cyclic $p$ -blocks, Linckelmann established that the source algebra equivalence class is fully determined by the pair:

The Brauer tree $T$ of the block (a planar embedding with possible exceptional vertex and multiplicity),
An indecomposable endo-permutation $kD$ -module $W(B)$ (Linckelmann's invariant), encoded as a class in the Dade group $D_k(D)$ .

Concretely, $W(B)$ corresponds to a characteristic subset $A\subseteq\Lambda = \{1,2,\ldots,l-1\}$ for $D \cong C_{p^l}$ , classified by the sign-sequence of a non-exceptional ordinary character of $B$ on powers of a generator of $D$ . The precise algebra class (up to $D$ -interior equivalence) of the source algebra $A(B)$ is thus uniquely specified by $(T, A)$ (Hiss et al., 2022, Hiss et al., 13 Feb 2025, Hiss et al., 4 Jan 2026). The entire classification for central quotients of $SL_n(q)$ and $SU_n(q)$ is completed in (Hiss et al., 4 Jan 2026), showing that in the linear and in the $p \equiv 1\ (\mathrm{mod}\ 4)$ unitary cases the invariant is trivial, and providing explicit combinatorial families of $A$ for the $p \equiv -1\ (\mathrm{mod}\ 4)$ unitary case.

3. Classification Theorems and Reduction to Linear/Unitary Cases

The main theorem sets:

Theorem: The source-algebra equivalence class of a cyclic $p$ -block $B$ with defect group $D \cong C_{p^l}$ is determined by its Brauer tree (planar embedding and exceptional multiplicity) and the subset $A \subseteq \{1, \ldots, l-1\}$ corresponding to $W_D(A)$ .
For quasisimple classical groups (types $B$ , $C$ , $D$ ), every indecomposable $W(B)$ arises already from a block of a central quotient of a special linear or unitary group (Hiss et al., 13 Feb 2025, Hiss et al., 4 Jan 2026).

A key feature is that for $SL_n(q)$ all cyclic blocks are source-algebra equivalent to the case with trivial $W(B) \cong k$ , while in the unitary case with $p\equiv-1$ ~mod~4, the possible non-trivial invariants are explicitly tabulated—intervals, complements of singleton sets in intervals, and unions thereof.

Group Type	Invariant $W(B)$	Description (for $D \cong C_{p^l}$ )
$SL_n(q)$ , $SU_n(q)$ , $p\equiv1\!\pmod 4$	$W_D(\emptyset) \cong k$	Trivial class
$SU_n(q)$ , $p\equiv-1\!\pmod 4$	$W_D(A)$ , $A$ interval types	As in families (i)-(iii) in (Hiss et al., 4 Jan 2026)

This reduction makes the global structure of source-algebra equivalence computable for all cyclic blocks of classical finite groups (Hiss et al., 13 Feb 2025, Hiss et al., 4 Jan 2026).

4. Source Algebra Equivalence in Functional Program Semantics

In program semantics, source algebra equivalence appears as the algebraic equivalence relation generated by a set of equations or rewriting rules over a free term monad, commonly in the context of effectful functional programming (Voorneveld, 2020). The equivalence relation induced by a set of source equations $A$ on terms $T_\Sigma(N)$ (where $\Sigma$ is a signature) is defined as the smallest reflexive, transitive, substitution- and composition-closed, base-valued (ground-complete), and $\omega$ -admissible preorder containing $A$ and the induced order.

Equivalently, such an equational theory $I$ admits a canonical Eilenberg–Moore algebra structure $\alpha: T_\Sigma(V) \to V$ whose test preorder $\sqsubseteq_\alpha$ exactly coincides with $I$ ; i.e., two programs are source algebra equivalent if and only if they cannot be distinguished by any test derived from $\alpha$ . The relation between source equations (axiomatization), value models (EM algebras), and logical modalities is formally encapsulated in a complementarity theorem (Voorneveld, 2020).

5. Verification and Checking of Source-Level Algebraic Equivalence

In verification, particularly for array- and loop-rich imperative or functional code, source algebra equivalence provides a formal method for proving that original and transformed programs are functionally identical. The approach in (0710.4689) is to construct for each C function an array data dependence graph (ADDG) that captures variable/operator structure and symbolic data dependencies, and then to apply a symbolic algorithm that synchronously traverses the two ADDGs. The sufficient condition for equivalence is that all matched paths through the operator trees have the same sequence of operators and identical dependence mappings from output to input. Algebraic transformations (exploiting associativity and commutativity) and various loop restructurings are handled by symbolically flattening and matching expression trees, without loop unrolling. If any discrepancy is found, the checker produces specific diagnostic feedback. This method is scalable for kernels with hundreds of statements and provides precise pinpointing of transformation errors (0710.4689).

6. Contexts and Limitations

In modular representation theory, source algebra equivalence is complete for cyclic defect cases but has more intricate invariants for non-cyclic defects; extension to blocks with general defect groups requires additional invariants beyond $(T, W(B))$ (Hiss et al., 2022). In program semantics, the equivalence induced by source algebras coincides with behavioral equivalence only under base-valuedness and (for Boolean modalities) single-valuedness assumptions. For program verification, the symbolic ADDG approach is limited to codes in static single assignment form, with affine loops, and non-pointer-based memory semantics (0710.4689).

7. Connections to Other Forms of Algebraic Equivalence

Algebraic equivalence in the sense of algebraic cycles (see (Achter et al., 2016)) is a distinct topic, concerned with the parameter spaces and equivalence relations of cycles on algebraic varieties. However, the unifying thread across all domains is the role of families of transformations or parameterizations (source equations, Brauer trees, etc.) that encode when two objects are "the same" up to source-level operations; in every context, a combination of combinatorial, algebraic, and representational invariants acts as the classifier for the induced equivalence. The completeness, computability, and diagnostic power of these equivalences are directly determined by the richness and tractability of their invariants.

References:

(Hiss et al., 2022): G. Hiss & C. Lassueur, "On the source algebra equivalence class of blocks with cyclic defect groups, I"
(Hiss et al., 13 Feb 2025): G. Hiss & C. Lassueur, "On the source algebra equivalence class of blocks with cyclic defect groups, II"
(Hiss et al., 4 Jan 2026): G. Hiss & C. Lassueur, "On the source algebra equivalence class of blocks with cyclic defect groups, III"
(0710.4689): "Functional Equivalence Checking for Verification of Algebraic Transformations on Array-Intensive Source Code"
(Voorneveld, 2020): "From Equations to Distinctions: Two Interpretations of Effectful Computations"
(Achter et al., 2016): "Parameter spaces for algebraic equivalence"