We introduce a path-theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher level generalisations over fields of arbitrary characteristic. Our first main result is a "super-strong linkage principle" which provides degree-wise upper bounds for graded decomposition numbers (this is new even in the case of symmetric groups). Next, we generalise the notion of homomorphisms between Weyl/Specht modules which are "generically" placed (within the associated alcove geometries) to cyclotomic Hecke and diagrammatic Cherednik algebras. Finally, we provide evidence for a higher-level analogue of the classical Lusztig conjecture over fields of sufficiently large characteristic.
arXiv:1706.07128v2 [math.RT] 3 May 2018Inspired by these results, we provide the first conjectural framework for calculating the (graded) decomposition numbers of cyclotomic Hecke algebras over fields of sufficiently large characteristic in Subsection 10.2. We verify these conjectures in the cases of (i) maximal finite parabolic orbits with k arbitrary -generalising all the results of [Kle97, CMT08, TT13a] (ii) = 2 or 3 with e > n and k arbitrary -generalising [HM15, Theorems B3 & B5] and [BS11, Section 9] (iii) = 2 and h = 1 and k arbitrary [CGM03, Section 8] (iv) k = C. For = 1 the conjecture was formulated by Andersen [And98, Section 5] and proved by Riche and Williamson in [RW16, Theorem 1.9]. Conjecture A. Let e > h , κ ∈ I be an h-admissible multicharge, and k be field of characteristic p h . The decomposition numbers of A h (n, θ, κ) are given by d λµ (t) = n λµ (t)for λ, µ ∈ P n (h) in the first ep-alcove and n λµ (t) the associated affine parabolic Kazhdan-Lusztig polynomial of type
DECOMPOSITION NUMBERS OF CYCLOTOMIC HECKE AND DIAGRAMMATIC CHEREDNIK ALGEBRAS 3Our treatment covers the quotient algebras Q ,h,n (κ) for e ∈ Z >0 uniformly alongside the algebras H n (κ) for e = ∞ (of more generally e > n). The alcove geometries controlling the latter family of algebras are seen as simple sub-cases of those controlling the former family of algebras. The following conjecture (a refinement of [KR11, Conjecture 7.3] which was debunked in [Wil14, Section 4.2]) is the corresponding simplification of Conjecture A.Conjecture B. Let e = ∞ or more generally e > n and suppose κ ∈ I has no repeated entries. Let k be field of characteristic p . The decomposition numbers of A(n, θ, κ) are given by d λµ (t) = n λµ (t)for λ, µ ∈ P n and n λµ (t) the associated parabolic Kazhdan-Lusztig polynomials of type A