🗺️ Map of Characters of Sp(2m) at the powers of Kostant's element

🏷️ Representation theory · characters of Lie groups 05E10 20G05 17B22 05A15

12 results ✅ 0 formalized in Lean ◐ 0 with a part in Lean 💻 10 checked by a program written proof only (a result can carry several tags)

Dependencies in the written proof: an arrow A → B means that this proof of B uses A. Every result rests on its written proof; the tags record what was checked besides, and each statement's “Evidence & scope” block says how far. The colour of a box is its strongest tag, in the order above; the tags in the box give the rest. Boxes with a thick border are the main results; click a box to read the statement.

What the paper contributes

Kostant's element a_P is a point where every irreducible character of a compact simple group takes a value in {0, ±1}. The paper determines the irreducible characters of Sp(2m) at every power a_P^k.

When a character vanishes, and its value (Theorem A): it vanishes exactly when λ+ρ lies on more walls of the modular root arrangement than ρ does, equivalently when some folded residue class holds more entries than its capacity; otherwise the value is an explicit signed ratio of root products.
The absolute value is a product of dimensions of representations of GL, Sp and odd SO, times κ ∈ {1, 2} (Theorem B).
Every rational point of the same line: a non-vanishing character is, up to sign and that factor, such a product times a character of Sp(2s) or SO(2s), s ≤ m, at a regular element (Theorem 10.1); the points where the closed form holds for all weights are listed (Theorem C), and at the points where all characters are integers the remaining factor is 1, 2, 3, 4 or 6 (Theorem D).

Known and new, as the paper states it. For k prime to 2m+2, Theorem A is due to Cellini, Möseneder Frajria and Papi; for gcd(k, 2m+2) odd, Theorems A and B follow from Albion's factorisations of universal characters, with the sign added. New: for gcd(k, 2m+2) even and q ≥ 3, the values on the classes not conjugate to that of ρ and the factor κ; the list of points in Theorem C and the exclusion of the others; in Theorem D, the constants for q ≥ 4 even with q | 2m, q ∤ m, and for q = 6 with m ≡ 4 (mod 6).

Key tool: an identity of residue multisets (Proposition 5.1), which takes the place of the conjugacy hypothesis in the work of Nadimpalli, Pattanayak and Prasad.

🚀 Start here: Theorem A: when the character vanishes, and its value.

Main results:

Theorem A: when the character vanishes, and its value
Theorem B: |value| = κ × product of dimensions
Theorem C: where the closed form holds for all weights
Theorem D: the constant c ∈ {1,2,3,4,6}

🧭 Fields: where the paper lives and what it connects

Tools taken from other branches

Root systems → walls of the modular root arrangement: the count criterion, proved by the argument of Nadimpalli–Pattanayak–Prasad (the alcove argument of Cellini–Möseneder Frajria–Papi needs ρ regular and no longer applies) — Proposition 3.1, Theorem A
Elementary number theory → residues modulo q in folded classes; an identity of residue multisets — Corollary 4.1, Proposition 5.1
Algebraic combinatorics → partitions, Weyl dimension formulas; the factorisations of universal and classical characters (Albion, Ayyer–Kumari) — Theorem B
Algebraic number theory → Galois action on the spectrum: which points give integer characters (a lemma of Gannon–Schopieray) — Proposition 8.1
Enumeration → counting the non-vanishing residues — Theorem 9.1

What the results serve

Characters at elements of finite order ← Kostant's {0, ±1} theorem, extended to every power of a_P; the question of Nadimpalli, Pattanayak and Prasad on vanishing and values at torsion elements, addressed for these elements of Sp(2m) — Theorem A, Theorem B
Reeder's constants ← their values on the classes not conjugate to that of ρ — Theorem A, Proposition 5.1
Factorisations of classical characters ← a factor κ and a residual character of type C or D at every rational point of the line — Theorem B, Theorem 10.1, Theorem D