Pith Number

pith:PE4226RE

pith:2026:PE4226REANOT64JAKAGK35RG7C

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Alperin's Main Problem of Block Theory

Alexander Moret\'o

The sets of irreducible characters nonvanishing at an element x, together with the subnormalizer of x, are the right local objects for governing character values in finite groups.

arxiv:2605.11988 v2 · 2026-05-12 · math.RT · math.GR

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

The right local objects governing character values are not, in general, the sets Irr_{p'}(G) and the normalizers of Sylow p-subgroups, but rather the sets Irr^x(G) of irreducible characters not vanishing at a given element x, together with the subnormalizer subgroup Sub_G(x).

C2weakest assumption

That the newly defined sets Irr^x(G) and subgroups Sub_G(x) actually control character values in the manner required by the stated conjectures, an assumption that is only verified in selected families rather than derived from prior established results.

C3one line summary

The paper introduces a conjectural framework for Alperin's Main Problem using Irr^x(G) character sets non-vanishing at given elements and Sub_G(x) subnormalizers, recovering McKay's conjecture and verifying the main statements for simple groups with TI Sylow p-subgroups.

Cited by

1 paper in Pith

Subnormalizers and character correspondences in $p$-solvable groups

Receipt and verification

First computed	2026-05-22T01:04:06.104354Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

7939ad7a24035d3f7120500cadf626f8b43c2a264728fd097e01b3da31bf7664

Aliases

arxiv: 2605.11988 · arxiv_version: 2605.11988v2 · doi: 10.48550/arxiv.2605.11988 · pith_short_12: PE4226REANOT · pith_short_16: PE4226REANOT64JA · pith_short_8: PE4226RE

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/PE4226REANOT64JAKAGK35RG7C \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 7939ad7a24035d3f7120500cadf626f8b43c2a264728fd097e01b3da31bf7664

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "0b01000c3174e09f03d5aaefdb5b37bfb8c72123cbf21126cff7483e86e59606",
    "cross_cats_sorted": [
      "math.GR"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.RT",
    "submitted_at": "2026-05-12T11:39:18Z",
    "title_canon_sha256": "30e52ec84c1515396f9c04a476881e57d507770571b7bbc3b5ae54b502e5b28c"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.11988",
    "kind": "arxiv",
    "version": 2
  }
}