The $p$-Canonical Basis for Hecke Algebras

Tilting modules and the p-canonical basis

“…Consider the injective morphism ζ of [S3,Proof of Proposition 3.4]. Then using [JW,Lemma 4.3] it is not difficult to check that, for w P f W , the element p H w0w belongs to the image of ζ; then we can define p M w by the property that ζp p M w q " p H w0w . One can also define the "dual" basis of Hom Zrv,v ´1s pM sph , Zrv, v ´1sq as in [S3], so that one can at least make sense of the p-analogues of all the ingredients in [S3, Theorem 5.1].…”

“…If s and s 0 are the unique elements in S f and S S f respectively, one can easily check that pTps 0 ss 0 s ‚ 0q : ∇ps 0 s ‚ 0qq " 1. On the other hand, the coefficient of N s0ss0s on N s0s is 0, while 3 n s0s,s0ss0s " 1 (see [JW,§5.3]). Of course, s 0 ss 0 s ‚ 0 does not satisfy the condition in (1.7).…”

The anti-spherical category

“…This lead to a more effective means of calculating the p-canonical basis. The basic idea is that via localisation one can reduce calculations of the p-canonical basis in the anti-spherical module (which can be performed via diagrammatics, as explained in [JW15]) to certain linear algebra problems over a polynomial ring in one variable (the ring denoted R I in § 3.7). This algorithm has been further developed and implemented by the second author [Wil] to provide a powerful new means to calculate decomposition numbers for symmetric groups.…”