Maximal ideals in the finitary symmetric group algebra in characteristic two

Abstract: In 1996, Baranov and Kleshchev classified maximal ideals in the finitary symmetric group algebra over fields of characteristic not 2. The corresponding question in characteristic~2 has remained open since. In the current paper we solve the problem by establishing part of a conjectural connection between prime ideals in this group algebra and the recently defined higher Verlinde categories. To achieve this we investigate thick tensor ideals of tilting modules of the general linear group in characteristic 2.

• The paper proves that, in characteristic two, the finitary symmetric group algebra has exactly two maximal ideals: the augmentation ideal and the ideal generated by the trivial-representation idempotent of S₃.
• The paper identifies the second ideal through the degree-3 Jones–Wenzl idempotent in the infinite Temperley–Lieb quotient and shows it is not the radical of any trace functional.
• The paper uses higher Verlinde categories, tilting-module tensor ideals, and an iterated SL₂-product embedding to establish the classification and prove that the finitary alternating group algebra also has exactly two maximal ideals.

Background and problem

The finitary symmetric group $S_\infty = \bigcup_n S_n$ has a group algebra $\Bbbk S_\infty$ whose ideal structure depends sharply on the characteristic of the field $\Bbbk$. In characteristic zero there are exactly two maximal ideals — the annihilators of the trivial and sign representations [FL]. Baranov and Kleshchev showed that for characteristic $p>2$ there are precisely $p-1$ maximal ideals, classified via inductive systems, branching rules and the Mullineux involution. The case $p=2$ resisted classification for thirty years because the Mullineux involution is unavailable there, and the naive analogue of the odd-characteristic answer is false: prior work already exhibited two maximal ideals. The paper under review settles this open problem.

Theorem A: when $\mathrm{char}(\Bbbk)=2$, $\Bbbk S_\infty$ has precisely two maximal ideals: the augmentation ideal $I[1]$, and the ideal $I[2]$ generated by the primitive idempotent of the trivial representation of $\Bbbk S_\infty$0.

A useful complementary description runs through Markov traces. For $\Bbbk S_\infty$1, the functional $\Bbbk S_\infty$2 with $\Bbbk S_\infty$3 has radical $\Bbbk S_\infty$4, and for $\Bbbk S_\infty$5 the radicals $\Bbbk S_\infty$6 exhaust the maximal ideals; for $\Bbbk S_\infty$7, $\Bbbk S_\infty$8 is generated by the symmetriser of $\Bbbk S_\infty$9 and the skew symmetriser of $\Bbbk$0. For $\Bbbk$1, the second maximal ideal arises inside the infinite Temperley–Lieb quotient $\Bbbk$2 as the ideal generated by the Jones–Wenzl idempotent in degree $\Bbbk$3 (its inverse image in $\Bbbk$4 being $\Bbbk$5). Notably, this ideal is not the radical of any trace functional: its quotient is a direct limit of matrix algebras $\Bbbk$6 with block diagonal embeddings, which admits only the zero trace. This distinguishes it structurally from all maximal ideals in characteristics other than 2.

Inductive systems and the Verlinde connection

Maximal ideals correspond bijectively to minimal inductive systems (coherent collections of simples $\Bbbk$7 across all $\Bbbk$8), and prime ideals relate to T-indecomposable systems via thick tensor ideals in the universal category $\Bbbk$9. A conjecture of Coulembier predicts a bijection between T-indecomposable inductive systems and isomorphism classes of semisimple objects in the higher Verlinde category $p>2$0. Since Baranov–Kleshchev showed that every minimal system must be one of the candidates $p>2$1, Theorem A follows from:

Theorem B: for $p>2$2, each $p>2$3 is realised as the inductive system of a semisimple object in $p>2$4.

The proof strategy is thus to establish "enough" of the conjecture rather than classify branching behaviour representation by representation, as was done for $p>2$5.

Tensor ideals of tilting modules in characteristic two

The technical core concerns thick tensor ideals in $p>2$6 at $p>2$7. When $p>2$8, such questions reduce via Andersen's work and Gelfand–Kazhdan to a principal $p>2$9-subgroup; no principal $p-1$0 exists for $p-1$1 in characteristic 2. The paper constructs a replacement: an iterated embedding $p-1$2, where $p-1$3 is the position of the leading bit in the binary expansion of $p-1$4. Pulling back the prime ideal $p-1$5 of $p-1$6 — equivalently, taking the kernel of a tensor functor $p-1$7 sending $p-1$8 to an explicit semisimple object $p-1$9 — yields a prime thick tensor ideal $p=2$0.

The central result describes membership in $p=2$1:

Theorem (cell of the Steinberg module): for $p=2$2, $p=2$3 if and only if $p=2$4 with $p=2$5 (successive parts differ by 1 or 2) and $p=2$6 (partitions whose transpose is a sum of distinct $p=2$7, $p=2$8 ranging over the support of $p=2$9).

This rests on a four-way characterisation of negligible tilting modules (equivalence of negligibility, the combinatorial condition on weights, vanishing of the restricted character $\mathrm{char}(\Bbbk)=2$0 in the ideal $\mathrm{char}(\Bbbk)=2$1 of the Grothendieck ring of $\mathrm{char}(\Bbbk)=2$2, and the condition $\mathrm{char}(\Bbbk)=2$3), proved using Donkin's tensor product theorem, splitting properties of $\mathrm{char}(\Bbbk)=2$4, and a root-of-unity evaluation argument showing that characters of non-negligible modules cannot vanish at $\mathrm{char}(\Bbbk)=2$5. As a corollary, the cell of the first Steinberg module $\mathrm{char}(\Bbbk)=2$6 in $\mathrm{char}(\Bbbk)=2$7 consists exactly of the $\mathrm{char}(\Bbbk)=2$8 with $\mathrm{char}(\Bbbk)=2$9 and $\Bbbk S_\infty$0 — a characteristic-2 analogue of the classical cell theory that required genuinely new constructions.

From this, the paper derives $\Bbbk S_\infty$1, together with the concrete inclusion statements that drive minimality: $\Bbbk S_\infty$2 and $\Bbbk S_\infty$3 whenever $\Bbbk S_\infty$4 is obtained from the binary expansion of $\Bbbk S_\infty$5 by deleting at least one term. These two facts show $\Bbbk S_\infty$6 is not minimal for any $\Bbbk S_\infty$7, completing Theorem A. The same machinery gives Theorem (alternating groups): $\Bbbk S_\infty$8 also has exactly two maximal ideals, the intersections with those of $\Bbbk S_\infty$9 — a sharper contrast than for $I[1]$0, where $I[1]$1 has $I[1]$2 maximal ideals.

For Hecke algebras $I[1]$3 at parameter $I[1]$4 of order $I[1]$5, the paper records that there are $I[1]$6 maximal ideals when $I[1]$7 (copying Baranov–Kleshchev verbatim via Brundan's branching rules) and exactly one when $I[1]$8, $I[1]$9; whether positive-characteristic $I[2]$0 Hecke algebras have precisely two maximal ideals is left as a conjecture, since the Hopf-algebra methods used for $I[2]$1 do not extend naively.

Reductions to rigid categories and further progress

Two structural theorems connect tensor ideals in $I[2]$2 with better-studied rigid categories. First, prime thick tensor ideals not containing $I[2]$3 are preimages under restriction $I[2]$4; combined with classifications of thick tensor ideals in $I[2]$5 (one chain indexed by Steinberg modules) and in $I[2]$6 — where the paper extends Andersen's Lemma by removing the injectivity hypothesis, yielding a single chain for $I[2]$7 versus the interleaved chain with ideals $I[2]$8 for $I[2]$9 — this classifies all T-indecomposable inductive systems of length two and three for every $\Bbbk S_\infty$00. Second, an analogous reduction to the rigid envelope $\Bbbk S_\infty$01 (a quotient of the oriented Brauer category) holds even for tensor ideals, not just thick ones.

These reductions imply that every weakly T-prime ideal in $\Bbbk S_\infty$02 admits a categorical dimension, proving one conjecture outright, and establish substantial cases of the Verlinde bijection conjecture: all T-indecomposable systems of length $\Bbbk S_\infty$03 for all $\Bbbk S_\infty$04; all minimal systems $\Bbbk S_\infty$05 for $\Bbbk S_\infty$06; and, for $\Bbbk S_\infty$07, the systems $\Bbbk S_\infty$08 for $\Bbbk S_\infty$09, $\Bbbk S_\infty$10, and $\Bbbk S_\infty$11, each identified explicitly with $\Bbbk S_\infty$12 for a semisimple $\Bbbk S_\infty$13 (e.g. $\Bbbk S_\infty$14).

Limitations and open questions

The full Conjecture 5.1.2 of [Tprime] — a bijection between all T-indecomposable inductive systems and semisimple objects of $\Bbbk S_\infty$15 — remains open for general lengths beyond those listed above, and its resolution for $\Bbbk S_\infty$16 beyond minimal systems is not claimed. The classification of thick tensor ideals in $\Bbbk S_\infty$17 in characteristic 2 is not undertaken; the embedding $\Bbbk S_\infty$18 is motivated by analogy with the conjectural nilpotent-orbit classification of [AHR2], but only the Steinberg-cell application is carried out. The positive-characteristic $\Bbbk S_\infty$19 Hecke algebra question stands as an explicit conjecture. Finally, the proof of Lemma (Andersen upgrade) relies on Hypothesis (indecomposability of $\Bbbk S_\infty$20), known when $\Bbbk S_\infty$21; extending the $\Bbbk S_\infty$22 analysis to larger groups at small $\Bbbk S_\infty$23 would require verifying this hypothesis independently.

Conclusion

The paper resolves a thirty-year-old classification problem by proving that $\Bbbk S_\infty$24 and $\Bbbk S_\infty$25 each have exactly two maximal ideals in characteristic 2, and identifies the second maximal ideal concretely via the degree-$\Bbbk S_\infty$26 Jones–Wenzl idempotent in the Temperley–Lieb quotient. Methodologically, it demonstrates that higher Verlinde categories control the ideal structure of $\Bbbk S_\infty$27 in characteristic 2, develops new tools — the $\Bbbk S_\infty$28 replacement for the principal $\Bbbk S_\infty$29, a strengthened Andersen Lemma, and reductions from $\Bbbk S_\infty$30 to rigid categories — and verifies significant cases of the conjectural dictionary between T-prime ideals and $\Bbbk S_\infty$31.