Split semisimplicity separates the Jucys–Murphy eigenvalues

On the semisimple Hecke algebra Hₙ(q) with Q = q² the Jucys–Murphy elements act on the new box of a one-box branching μ ⊂ μ + c by the eigenvalue Q ^ content c, and YoungDiagram.injOn_zpow_cellContent_iff says that these eigenvalues separate the addable cells of every diagram of size less than n exactly when Q ^ d ≠ 1 for 2 ≤ d ≤ n. This file derives that condition from the split-semisimplicity criterion: over a commutative ring without zero divisors, it holds whenever Q ≠ 1 and the product ∏_{d=1}^{n} [d]_Q of Q-integers is nonzero.

Main results

• YoungDiagram.injOn_zpow_cellContent_of_prod_geom_sum_ne_zero: without zero divisors, the split-semisimplicity criterion ∏_{d=1}^{n} [d]_Q ≠ 0 together with Q ≠ 1 makes the powers Q ^ content c separate the addable cells of every diagram of size less than n.

References

• A. Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. 75 (1997), 99–133, for the Hecke eigenvalues Q ^ content.

theorem YoungDiagram.injOn_zpow_cellContent_of_prod_geom_sum_ne_zero {μ : YoungDiagram} {R : Type u_1} [CommRing R] [NoZeroDivisors R] {Q : Rˣ} {n : ℕ} (hQ : ↑Q ≠ 1) (hprod : ∏ d ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range d, ↑Q ^ i ≠ 0) (hμ : μ.card < n) : Set.InjOn (fun (c : ℕ × ℕ) => ↑(Q ^ cellContent c)) {c : ℕ × ℕ | μ.IsAddable c}

Split semisimplicity separates the Jucys–Murphy eigenvalues. Over a commutative ring without zero divisors, if the unit Q is not 1 and the product ∏_{d=1}^{n} [d]_Q of the Q-integers [d]_Q = 1 + Q + ⋯ + Q ^ (d - 1) is nonzero, then on every Young diagram of size less than n the powers Q ^ content c of distinct addable cells are distinct.