Counterexamples to James's Conjecture

• The paper demonstrates that minimal explicit counterexamples to James's conjecture expose nontrivial adjustment matrices in small-weight blocks of symmetric groups and Hecke algebras.
• The study employs computational Gram matrix analysis and parity sheaf theory to link torsion in intersection cohomology with unexpected modular decomposition behavior.
• The findings underscore the need for refined invariants and geometric methods to understand the interaction between modular reduction and graded structures in modern representation theory.

James’s conjecture concerns the modular representation theory of symmetric groups and their Hecke algebras, asserting stability of decomposition matrices under transition from characteristic zero to sufficiently large positive characteristic in blocks of small defect (weight). Counterexamples, initially only of astronomical size, have been identified, demonstrating structural phenomena that invalidate the conjecture’s predicted triviality of adjustment matrices even in low-weight blocks. Recent advances have constructed minimal explicit counterexamples both for symmetric groups, their Iwahori–Hecke deformations, and for cyclotomic Khovanov–Lauda–Rouquier (KLR) algebras of Dynkin type, revealing the deep role of torsion in intersection cohomology and the non-trivial interaction of modular reduction with graded structures in modern representation theory.

1. Statement of James’s Conjecture and Its Blockwise Form

Let $\mathscr{H}_n(q)$ denote the Iwahori–Hecke algebra of type $A_{n-1}$ over an algebraically closed field $\mathbb{F}$ of characteristic $p \geq 0$, with quantum parameter $q \in \mathbb{F}^\times$. Its Specht modules $S(\lambda)$ are indexed by partitions $\lambda \vdash n$, while simple modules $D(\mu)$ are indexed by $e$-restricted partitions, where $e$ is the quantum characteristic defined by $A_{n-1}$0.

The decomposition matrix $A_{n-1}$1 records the composition multiplicities $A_{n-1}$2. Over characteristic zero, the decomposition is described by the LLT/Ariki theorem. James’s conjecture predicts that, for blocks of weight $A_{n-1}$3, the adjustment matrix $A_{n-1}$4 in the equation $A_{n-1}$5 is the identity. That is, modular decomposition numbers should coincide with the characteristic-zero ones for such blocks. The blockwise version specifically posits that in every block of weight $A_{n-1}$6, the adjustment matrix is trivial, i.e., $A_{n-1}$7 (Speyer, 13 Jan 2026, Williamson, 2012).

2. Counterexamples in Symmetric Groups and Hecke Algebras

Geordie Williamson’s work established the first counterexamples to James’s conjecture, using parity sheaves and intersection cohomology on flag varieties to demonstrate the existence of extra composition factors in high characteristic and large rank. The smallest counterexample arising from Williamson’s general method for symmetric groups appears for $A_{n-1}$8 and $A_{n-1}$9. However, these counterexamples remained computationally inaccessible for explicit detailed study (Speyer, 13 Jan 2026).

A new minimal explicit counterexample was constructed within the Iwahori–Hecke algebra $\mathbb{F}$0 for $\mathbb{F}$1, $\mathbb{F}$2 (with $\mathbb{F}$3 a primitive fourth root of unity), and $\mathbb{F}$4. The relevant principal block has 4-core $\mathbb{F}$5 and weight $\mathbb{F}$6, labelling simple modules by 4-restricted partitions of 24. The distinguished pair

$\mathbb{F}$7

both 4-restricted and in the principal block, exhibits the counterexample. Computation of graded decomposition numbers via a Gram-matrix argument in the cyclotomic KLR algebra model shows that, while in characteristic zero $\mathbb{F}$8, in characteristic $\mathbb{F}$9 one obtains $p \geq 0$0, introducing an extra simple summand in degree zero (Speyer, 13 Jan 2026).

3. The Mechanism of the Minimal Counterexample

The salient mechanism underlying this minimal counterexample is detection of the drop in rank of the Gram matrix modulo $p \geq 0$1. The homogeneous bilinear form on the graded Specht module $p \geq 0$2, modelled in the KLR algebra framework, permits identification of the integer Gram matrix $p \geq 0$3 acting on weight spaces corresponding to residue sequences. Computation in the case at hand yields a Smith normal form with elementary divisors $p \geq 0$4. Over $p \geq 0$5, the rank of $p \geq 0$6 drops by $p \geq 0$7, producing an extra direct summand $p \geq 0$8 at degree zero in the composition series of $p \geq 0$9, reflecting a jump in the graded decomposition multiplicity (Speyer, 13 Jan 2026).

The partial graded decomposition matrices for this block (principal block of $q \in \mathbb{F}^\times$0 at $q \in \mathbb{F}^\times$1), focusing on the distinguished pair, are:

Matrix	$q \in \mathbb{F}^\times$2	All Other $q \in \mathbb{F}^\times$3
$q \in \mathbb{F}^\times$4 (char 0)	$q \in \mathbb{F}^\times$5	agree with James’s conjecture
$q \in \mathbb{F}^\times$6 (char 7)	$q \in \mathbb{F}^\times$7	agree with James’s conjecture

Therefore, the graded adjustment matrix $q \in \mathbb{F}^\times$8 differs from the identity only at the $q \in \mathbb{F}^\times$9 position, with $S(\lambda)$0, displaying a nontrivial correction to the conjectured identity matrix.

4. Analogue in Khovanov–Lauda–Rouquier Algebras

James’s conjecture has a parallel (“Kleshchev–Ram conjecture”) in the context of graded cyclotomic KLR algebras associated to simply-laced Dynkin diagrams, motivated by the categorification paradigm and isomorphisms relating group/Hecke algebras to cyclotomic quotients. The conjecture postulates that, once the characteristic exceeds a specific lower bound, decomposition matrices should again be trivial (i.e., identity).

A minimal counterexample for KLR algebras arises in type $S(\lambda)$1, characteristic $S(\lambda)$2, with a specific dimension vector for the associated quiver. The mechanism is geometric: the failure of parity sheaves to remain indecomposable after extension to characteristic zero, traced to the presence of $S(\lambda)$3-torsion in the cohomology of intersection complexes on a quiver variety stratified by orbits labelled by two particular modules ($S(\lambda)$4 and $S(\lambda)$5). The failure is locally modelled by the Kashiwara–Saito singularity, which ensures that the decomposition matrix is non-identity in this case (Williamson, 2012).

Extensions of this phenomenon to all primes in type $S(\lambda)$6 quivers are secured via flag varieties whose intersection cohomology carries $S(\lambda)$7-torsion in some costalk, demonstrating the systemically nontrivial adjustment matrices and universal failure of the conjecture in type $S(\lambda)$8 for all characteristics (Williamson, 2012).

5. Methodological Approaches

Explicit detection of the minimal counterexample in the Hecke algebra setting is enabled by:

• Realization of the relevant block as a cyclotomic KLR algebra (type $S(\lambda)$9), giving access to natural $\lambda \vdash n$0-gradings on cell modules.
• Calculation of Gram matrices for graded Specht modules on selected residue-sequence weight spaces, utilizing the Hu–Mathas cellular basis.
• Implementation of the LLT/Ariki algorithm in computer algebra systems (GAP) for characteristic zero graded decomposition matrices.
• Cross-verification via Smith normal form of integral Gram matrices to detect rank drops mod $\lambda \vdash n$1 (Speyer, 13 Jan 2026).

In the KLR context, geometric and cohomological tools, including parity sheaves, intersection cohomology, and analysis of orbit stratifications for quiver varieties, are critical for linking modular representation phenomena to the (non-)triviality of decomposition matrices (Williamson, 2012).

6. Minimality and Structural Implications

No blocks for $\lambda \vdash n$2, $\lambda \vdash n$3, or for $\lambda \vdash n$4, up to comparably large $\lambda \vdash n$5, exhibit a violation of James’s conjecture of the above type. The rank $\lambda \vdash n$6 at $\lambda \vdash n$7, $\lambda \vdash n$8, is minimal among known explicit counterexamples for $\lambda \vdash n$9. For $D(\mu)$0, the pattern of failure is different and requires separate treatment (Speyer, 13 Jan 2026).

These counterexamples highlight that the presence of torsion in the cohomology of intersection complexes coincides with the appearance of nontrivial adjustment matrix entries, undermining the naïve expectation of decomposition stability in small-weight blocks. The identification of the Kashiwara–Saito singularity as the locus for such failures in quiver settings connects geometric singularities to modular phenomena in categorification (Williamson, 2012).

7. Impact and Future Research Directions

The discovery of explicit and minimal counterexamples to James’s conjecture demonstrates that even the most optimistic stabilization scenarios for decomposition numbers in the modular representation theory of symmetric groups, Hecke, and KLR algebras fail in small-weight blocks and moderate rank. This underscores the necessity for refined invariants and the study of torsion-based obstructions in geometric representation theory.

Future investigations may focus on characterizing the precise classes of blocks and parameters where decomposition-triviality persists, advances in parity-sheaf theory, further exploitation of Soergel bimodule machinery, and geometric analysis of quiver and flag varieties to predict or classify torsion phenomena responsible for the breakdown of naive adjustment-matrix predictions (Speyer, 13 Jan 2026, Williamson, 2012).