The p-canonical basis for Hecke algebras and p-cells
Lars Thorge Jensen
Max Planck Institute for Mathematics
November 22, 2017
Motivation
Motivation
Notation: k =k field of characteristicp>0.
Long-standing open problems in modular representation theory (forp>0):
What are the characters of ...
I modular irreducible modules ofSr overk forp6r?
I indecomposable tilting modules ofGLn overk?
The following basis contains the answer to these questions...
Thep-canonical basis
Idea for the p-canonical basis
Notation (forG⊇B⊇T a split, sc alg. group/kwith Borel and max. torus):
I the affine Weyl groupW :=Wfn ZΦ as a Coxeter system (W,S),
I kHthe Hecke category (defined over k of characteristicp),
I Hthe Hecke algebra assoc. to (W,S) overZ[v,v−1] .
Theorem (Elias-Williamson, Soergel, Kazhdan-Lusztig, . . . )
There exists an isomorphism ofZ[v,v−1]-algebras:
ch : [kH]−→ H, [Bs]7−→Hs for s∈S where[kH]denotes the split Grothendieck group of kH.
Definition
Thep-canonical basisofH is given by:
Thep-canonical basis
Properties of the p-canonical basis
Instead of precisely stating its properties, we give the following slogans:
I Thep-canonical basis is a positive characteristic analogue of the Kazhdan-Lusztig basis.
I Thep-canonical basis loses many of thecombinatorial propertiesof the KL basis, but preserves itspositivity properties(as stated in the Kazhdan-Lusztig positivity conjectures).
I The KL-basis (and the KL-polynomials) are ubiquitous in representation theory (e.g. in theKL-conjecturesrelating characters of Verma and simple modules for a semisimple Lie algebra), thep-canonical basis is expected to play a similar role inmodular representation theory.
Back to motivation
p-Canonical basis in type A
g1for p = 3
3Hs =Hs
3Hst = Hst
3Hsts = Hsts
3Hstst = Hst +Hstst
3Hststs =Hs + Hststs
3Hststst = Hststst
3Hstststs = Hststs + Hstststs
3Hstststst= Hstst + Hstststst
Figure:The 3-canonical basis in terms of the Kazhdan-Lusztig basis
0 0
1
1 2
2 3
3 4
4 5
5
6
6
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7
8
8
9
9
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10
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25
25 n
m
Figure: The multiplicities of ∆(m) in T(n) forp= 3
Back to motivation
p-Cells
p-Cells give a first approximation of the multiplication in thep-canonical basis.
Definition
Define a pre-order 6p
R onW via:
x
p
6R
y⇔ ∃h∈ H:pHx occurs with non-zero coefficient in pHyh The equivalence classes w.r.t. 6p
R are called right p-cells. The leftp-cell (resp.
two-sided)p-cell preorder 6p
L (resp. 6p
LR) as well as left (resp. two-sided) p-cells are defined similarly.
Back to motivation
Right p-cells in type A
g2and p = 5
••
••••
Back to motivation
p-Cells in finite type A
In finite typeAn+1, we can explicitly describep-cells via the
Robinson-Schensted correspondence which establishes a bijection between the symmetric groupSn and pairs of standard tableaux withn boxes mapping w∈Sn to (P(w),Q(w)). Following Ariki’s work we can prove:
Theorem
For x,y∈Sn we have:
x ∼p
L y⇔Q(x) =Q(y), x ∼p
R y⇔P(x) =P(y), x ∼p
LR y⇔Q(x)and Q(y)have the same shape.
In particular, Kazhdan-Lusztig cells and p-cells of Sn coincide.
References
References I
Susumu Ariki,
Robinson-Schensted correspondence and left cells
Combinatorial methods in representation theory (Kyoto, 1998) Adv. Stud. Pure Math., vol. 28, Math. Soc. Japan, Tokyo, 2000, pp. 1–20.
Henning Haahr Andersen,
Cells in affine Weyl groups and tilting modules
Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math., vol. 40, Math. Soc. Japan, Tokyo, 2004, pp. 1–16.
Lars Thorge Jensen and Geordie Williamson, The p-Canonical Basis for Hecke Algebras
to appear in Perspectives on Categorification, Contemp. Math., Amer. Math. Soc..
David Kazhdan and George Lusztig,
Representations of Coxeter groups and Hecke algebras.
Invent. Math.53, 1979, no. 2, 165–184.
,
Cells in affine Weyl groups.
