p-Kazhdan-Lusztig Polynomials Unify Modular Characters and Tilting Multiplicities — E8 Intelligence Research

Abstract

FINDING: The p-Kazhdan-Lusztig polynomials govern modular character formulas for reductive groups in positive characteristic, linking perverse sheaves on affine flag varieties to tilting module multiplicities. | MATH: The key object is the p-Kazhdan-Lusztig basis { underlineC_w } of the affine Hecke algebra over ℤ, specializing at v ↦ 1 with p-dependent coefficients. The character formula (Williamson–Riche) states: ch(L(λ)) = ∑_{μ ≤ λ} (−1)^{ℓ(λ)−ℓ(μ)} P_μ,λ^(p)(1) ch(Δ(μ)), where P_μ,λ^(p) are p-KL polynomials. These satisfy P_μ,λ^(p) = ∑_i a_i vⁱ with a_i ∈ ℤ_{≥ 0}, and crucially P_μ,λ^(p) ≠ P_μ,λ (classical KL) for small p. The affine flag variety G((t))/I (Iwahori) has perverse sheaves whose stalks compute these polynomials; the modular case requires 𝔽_p-coefficient

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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BibTeX RIS

BibTeX

@misc{caldin2026pkazhdanlusztig,
  title = {{p-Kazhdan-Lusztig Polynomials Unify Modular Characters and Tilting Multiplicities — E8 Intelligence Research}},
  author = {Andrew Stewart Caldin},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23635194},
  url = {https://zenodo.org/records/23635194},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - p-Kazhdan-Lusztig Polynomials Unify Modular Characters and Tilting Multiplicities — E8 Intelligence Research
AU  - Andrew Stewart Caldin
PY  - 2026
DA  - 2026/10/25
PB  - Zenodo
DO  - 10.5281/zenodo.23635194
UR  - https://zenodo.org/records/23635194
AB  - FINDING: The p-Kazhdan-Lusztig polynomials govern modular character formulas for reductive groups in positive characteristic, linking perverse sheaves on affine flag varieties to tilting module multiplicities. | MATH: The key object is the p-Kazhdan-Lusztig basis { underlineC_w } of the affine Hecke algebra over ℤ, specializing at v ↦ 1 with p-dependent coefficients. The character formula (Williamson–Riche) states: ch(L(λ)) = ∑_μ ≤ λ (−1)^ℓ(λ)−ℓ(μ) P_μ,λ^(p)(1) ch(Δ(μ)), where P_μ,λ^(p) are p-KL polynomials. These satisfy P_μ,λ^(p) = ∑_i a_i v^i with a_i ∈ ℤ_≥ 0, and crucially P_μ,λ^(p) ≠ P_μ,λ (classical KL) for small p. The affine flag variety G((t))/I (Iwahori) has perverse sheaves whose stalks compute these polynomials; the modular case requires 𝔽_p-coefficient  Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER  -

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