Semisimple character partitions and a rational cycle obstruction

Definitions and mathematical statement

For a finite connected graph $G$, let $K(G)$ be its critical group, the finite cokernel of a reduced graph Laplacian; $|K(G)|=\tau(G)$, the spanning-tree number. Work with a nonloop nonbridge edge $e$, allowing parallel edges after contraction and ignoring loops in the critical-group Laplacian. Then \[ \tau(G)=\tau(G\setminus e)+\tau(G/e). \] The source asks for a meaningful deletion--contraction recurrence, preferably an algebra-level exact sequence, reflecting more than these orders. It invokes Merino's relation between the Tutte polynomial and chip-firing, suggests beginning with cycles, and asks for an analogue of an unspecified hyperplane-arrangement algebra sequence.

The supplied report says Manjunath's 2022 critical-module sequences give a substantial positive answer in a module setting, without identifying a complete theorem or asserting that they are literal critical-group-algebra sequences. For the latter formulation it proves the following semisimple statement. Let $k$ be algebraically closed of characteristic zero and $\widehat K=\operatorname{Hom}(K,k^*)$. Fourier evaluation gives $k[K]\simeq k^{\widehat K}$. Consequently a sequence \[ 0\longrightarrow k[K(G/e)]\xrightarrow{\iota}k[K(G)] \xrightarrow{\pi}k[K(G\setminus e)]\longrightarrow0 \] exists with $\iota$ identifying the left algebra with an ideal, and is split. The ideal inclusion is not required to preserve the ambient identity; requiring every displayed map to be unital would be incompatible with a proper nonzero ideal.

Such sequences are exactly partitions $\widehat{K(G)}=X_C\sqcup X_D$, together with bijections $X_C\simeq\widehat{K(G/e)}$ and $X_D\simeq\widehat{K(G\setminus e)}$. Inclusion is extension by zero on the ideal coordinates and quotient is restriction to the complementary coordinates. Thus existence follows from the additive spanning-tree count and by itself carries no prescribed group structure or naturality. For a specified symmetry group acting on the three character sets, equivariance requires and is equivalent to an invariant partition and equivariant bijections; this is not automatically a functorial construction for all graph morphisms.

For a cycle $C_m$, $m\geq3$, deletion gives a tree and contraction gives $C_{m-1}$, with the length-two cycle interpreted as parallel edges. Hence $K(C_m)\simeq{\mathbb{Z}}/m{\mathbb{Z}}$ and the sequence has endpoint dimensions $m-1,1$. If $g$ generates ${\mathbb{Z}}/m{\mathbb{Z}}$ and $\zeta_m$ is a primitive root, set \[ e_j=\frac1m\sum_{r=0}^{m-1}\zeta_m^{-jr}g^r\qquad(0\leq j<m). \] The augmentation quotient has kernel $\bigoplus_{j=1}^{m-1}ke_j$; any labeling of these primitive idempotents by the characters of ${\mathbb{Z}}/(m-1){\mathbb{Z}}$ gives the reported explicit ideal injection. The same applies to a unicyclic graph with trees attached when $e$ is on its cycle. These sequences are not induced by an injection of the endpoint critical groups and are not Hopf-exact: in the cycle example the relevant group orders do not satisfy the required subgroup/divisibility structure.

The coefficient field matters. For the triangle, \[ {\mathbb{Q}}[K(C_3)]\simeq{\mathbb{Q}}\times{\mathbb{Q}}(\zeta_3),\qquad {\mathbb{Q}}[K(C_2)]\simeq{\mathbb{Q}}\times{\mathbb{Q}},\qquad {\mathbb{Q}}[K(\mathrm{tree})]={\mathbb{Q}}. \] The rank-two ideal is a quadratic field rather than ${\mathbb{Q}}^2$, and the rank-one quotient in the opposite endpoint order is again that quadratic field. Thus neither endpoint order gives an analogous ideal/quotient exact sequence over ${\mathbb{Q}}$. Any such sequence over ${\mathbb{Z}}$ would yield one over ${\mathbb{Q}}$ by tensoring, so is also excluded. This is an obstruction to the specified literal formulation, not to every module-level or differently normalized deletion--contraction theory.

Short English statement

Over an algebraically closed characteristic-zero field, the reported group-algebra sequences exist because they amount to arbitrary partitions of character coordinates. Their existence alone does not encode a natural critical-group recurrence. Cycles give explicit examples, while the triangle rules out the same ideal/quotient formulation over ${\mathbb{Q}}$ and ${\mathbb{Z}}$; separate critical-module results address a broader version.

Sources

• Local registry: permanent UnsolvedMath ID $20000975$, exact public code catalogue entry; aim-combinatorics-notes.json, notes[99].
• AIM workshop Generalizations of chip-firing and the critical group, Problem 10: https://aimath.org/pastworkshops/chipfiringproblems.pdf. The supplied verification reports printed page 5/PDF page 4, as given; no independent PDF access is claimed.
• Lionel Levine poses the recurrence and arrangement-sequence questions; Jeremy Martin suggests cycles. The source cites Criel Merino [25] without the bibliography entry and does not identify the hyperplane-arrangement algebra.
• Supplied research summary, "Semisimple character partitions and a rational cycle obstruction," attempt 1, refers to Manjunath's 2022 critical-module sequences without full publication details or theorem statement. Original remarks and literature fields are empty.