REV 2019 Day 8B.Bmbakere

Matrix groups over finitefieldsand their representationtheoryC

The earlierproblem 2 was secretly

motivatedby rep n theory

Sal Schurpolynomialsare characters ofrepins Vof Glu

f n xn invertiblematriceswith entries

Inthis project want tostudyrep1ns ofgroups like Glen gwhere Fg finitefieldwithof

elements

pk p aprime

Vaybriefmotivation

Theygive examplesoffinitesimplegroupsC no

nontrivialnormalsubgroups

classifiedCin thousands ofpagesfinishedaround2004 a s

cyclicgroups ofprime order

alternatinggroups Ah n 25

finitegroupsofLietype

26 more examples sporadic1860 s Mathieu 5

19 Griess Monstergwup

NumbertheoryDiophantine equationsf solutions tointegerrational

rational

coefficientequations

e g E yZx3tXH

easier EGFp txylyxkx.itmodp

crazyida make a generating

fee Fasano

acp madefrom EIDacn s madefrom aCpb

AmazingconjectureBirch SwinnertonDyer

gOrderofvanishingofLCEdats

r gives Cuplofniklymanysolutions

the ofsolutions to ECON

LKeown ECQ E finitegroup

I I s 12

Study L functionsfromDiophantine equations using

L functignstomrephtheory

madffomrephsotgroups

like Ancap1 go.imenaEeaiEsin

EiEs

theoyofGhdQp is

alotlikerephtheongotGLnCFp

whatgroupswill westudywhatdo weknowabout

rephtheory.tlBegin with GbCFp

2 2 matrices

withde toirFp

If G is finiteII dimpf lat

firedrepinofirredrophs ofG conjugacy

classes ofG

GEED013 1 57 Bsourbelgoup

FACT Bmhatdecomposition

Adt B w Bw Bwhere w GOT

disjointmion

and BWBfb.dz bybaEBdoublecoset

so can move totheotherBin the doubleasset

MoregenerallyGlen fewBwBwhere B If fFEffpW permutationmatrices

in Glu fSu symmetricgroup

EXERCISE I 8Rental

show GhettoFB UB Bb Show that for Gluffgif T Cams

cnormalizer of Tin G

then NatHT E Sn

W Weylgroup

c Compute Iadept

RENEXERUSEIIa Determine theconjugagdassesotGbat

Whatgroupswill we studyA Groups G withBordsubgroups

B 2T atoms for somedefn of Bsothat

G wBwBwtf.NET T

Which groupswarepossible

E AsubsetotthefiniteCoxetergroupsCoxetergroypsaregroupsgenerated byreflections

Theyharepresentationsgiven

by Coxeterdiagramsgraphswithvertices generating

reflection siedges Glabemled

Sjm

1

Hip kids don'twrite m if m 3S SE SzS S2

a3 7

and don'twrite the edge at allif m 2 SEES D

There are 4 infinitefamiliesofgraphs groups whose graph isconnected irreducible wherethe

groupendsup being finite

An l Esmee g AILS s2 sf sf 1

955 843E

S 7 7 1,2S2 1 7 G s

Bnkns s n

Dn 4 Inoatystallgogrnaoph

2

imsdihedrdgnapsgien.meF4 E6 Et EgGz 2,43

Westudygroups ofthis typehaving Bmhat decomposition GUBuB

and w fromtheabove listhow

There are nice presentationsofthesegroups using

rootsystemsacted on by W seeRehmann

reference

thegroups Gare reallyfixedpomtsoffwbeninsaclngonreduc.tn

algebraicgroupsdefnedorertthavea nice presentation in termsofW

and its associated wotsystem

RepintheayofsachGE.g.GLp

Pickalbig abdiansubgroupConsider 1 1007Itsiweduciblerepinsaretdine

FEET5 t

Kiki.E qx

XH XfaHdb

To get more representationsinduceup to G i e

try Indf XNotsogood.becauseitbreducibk.tnsteaddosomething trickier

thinkof X as rep n of

B HE 73 It bEB write b tu teFfFoEB

ne U Effigyand define X b XHI

Newda IndBG XThisturns out notobvioustobe an irreduciblerep not

dimension LGBT p thunless X Xzthen reducible get 1 dimiland a p dime

irreducible

Rt e9bAssumingtheabovefacts

about

IndBLX howmanyirreduciblerepins

of 0k are left

Upshot BeginwithcharactersottomsT

mimeographsMacdonald11968

conjectured there is a correspondencebetweencharactersottori andirreduciblerepinsofreductive splitalgebraicgroupsoverfg

there are more tori insideGLIEDand its relatives

Gf1 1 2DuffyFD Ep

ez atomswith E Fp B

ReupholsterStudytherephtheoryot

genlmlextusionsotGHFwhere GBasplitannected

reductivealgebraicgroup

e I zS GEq 71

Turnsout withseffGtobeafiniteabdgionp Ganterof

THM Steinberg's

Guff is trivialexceptfor 11exceptionsG simply

connected simple

algebraicgroupdefined overFg

e g Ay 4 StaffEAifeng.com

Atlas ofFiniteGroupRepins says

HI n 3 1015