Index of Notation

Index of Notation

General symbols

C field of complex numbers

F_� finite field with �elements

� ?

´1inC

IdM identity map on the setM

Imp�q image of the map �

kerp�q kernel of the morphism �

N the natural numbers without 0

N₀ the natural numbers with 0

P the prime numbers in Z

Q field of rational numbers

R field of real numbers

Z ring of integer numbers

Z_•��Z_°��Z_§��Z_†� t�PZ|�•� (resp. �°�� �•�� �†�qu

|X| cardinality of the set X

δ�� Kronecker’s delta

î union

≤ disjoint union

ì intersection

∞ summation symbol

±,ˆ cartesian product

‘ direct sum

b tensor product

H empty set

@ for all

D there exists

– isomorphism

� complex conjugate of �PC

�|�,�-� � divides�, � does not divide�

�|S restriction of the map� to the subsetS

Group theory

A� alternating group on� letters

C� cyclic group of order �in multiplicative notation

CGp�q centraliser of � inG

CpGq set of conjugacy classes ofG

D2� dihedral group of order 2�

FixXp�q set of fixed points of �on X

Skript zur Vorlesung: Charaktertheorie SS 2020 58

rG�Gs orG¹ commutator subgroup ofG

G{N quotient group G modulo N

GL�pKq general linear group overK

H§G, H†G H is a subgroup of G, resp. a proper subgroup

N úG N is a normal subgroupG

NGpHq normaliser of H inG

PGL�pKq projective linear group overK

Q8 quaternion group of order8

S� symmetric group on �letters

SL�pKq special linear group over K

Syl_�pGq set of Sylow�-subgroups of the groupG

ZpGq centre of the groupG

Z{�Z cyclic group of order� in additive notation

|G| order of the groupG

|G:H| index ofH inG

r�s conjugacy class of �

r�� �s commutator of� and �

x�y cyclic group generated by�

x�|�^�“1y cyclic group of order� generated by�

Rings and linear algebra

RrXs ring of polynomials in an indeterminateX over

the ringR

R^ˆ group of units of the ringR

charpKq characteristic of the fieldK

det determinant of a matrix/linear transformation

dimK K-dimension

EndKpVq endomorphism ring of theK-vector space V

GLpVq set of invertible linear transformations of

the vector spaceV

x�1�¨ ¨ ¨� ��yK K-linear span of the sett�1�¨ ¨ ¨� ��u

M�ˆ�pKq ring of�ˆ�-matrices with coefficients inK

M�pKq ring of�ˆ�-matrices with coefficients inK

K algebraic closure of the fieldK

Tr trace of a matrix/linear transformation

W §V W is aK-subspace ofV

t�1�¨ ¨ ¨ � ��u a basis of K^�

p�1�¨ ¨ ¨� ��q an ordered basis of K^�

Representations and characters

C1� � � � �C� the conjugacy classes ofG

Cx1� � � � �xC� the class sums ofG

��pGq C-vector space of class functions onG

Inf^G_G{N inflation fromG{N toG

Ind^G_H,Ò^G_H induction fromH toG

IrrpGq “ tχ1� � � � � χ�u set of irreducible characters ofG

kerpχq kernel of the characters of χ

�pG�Kq space of K-valued functions ofG

KG group algebra ofG over the fieldK

Skript zur Vorlesung: Charaktertheorie SS 2020 59 Res^G_H,Ó^G_H restriction fromG toH

ZpKGq center ofKG

Zpχq center of the character χ ρ„ρ¹ ρ is equivalent toρ¹

ρ_reg the regular representation ofG

ρV representation associated to the G-vector spaceV χ_reg regular character ofG

χ_V character of the G-vector spaceV ω1� � � � � ω� the central characters ofG x´�´yG scalar product on ��pGq 1G the trivial character ofG

Greek Alphabet

lower-case letter upper-case letter Name

α A alpha

β B beta

γ

gamma

δ

delta

ε, � E epsilon

ζ

zeta

η H eta

θ

theta

ι I iota

κ K kappa

λ

lambda

µ

M mu

ν N nu

ξ

xi

� O omicron

π� �

pi

ρ� � P rho

σ� ς

sigma

τ T tau

υ

upsilon

φ� �

phi

χ X chi

ψ

psi

ω

omega