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Physics 618 2020
• Fourier duality k, Pontryagin Duality
- Orthog .
Rel 's for Matrix elementsof imed . reps .
° Heisenberg Groups with no
canonical Lagrangian subgroups .
• Induced Representations
April 28,2020
£a§¥ttzimmee:Wnniiqueenneesssoff§*NNpeepof
#ei$¢$*§D
.
VV±LE$$§#s#$¢$"D==#¢$#$"$Ses
'
REIT:L!Ig .
(@Y*#*D=**$4fe$5
xef
But
£§EE§so
#ee#k§*§D¥=t#i$(§*§§fsSo there most be an intertwine ra:EE$d→E£§x&##*⇐m%Ssiµ*#'⇐¥*
Fourier Het
imnssmdj:{IEEE;÷§*****•
e.g. if s= IR,
£= R
Tuck )=§Eik×4c÷Realm) - Heists )
-sSx£:f , )k(GX , ),(5×d)=# ,
Z(Heis$×£D±U ).
Sx 's ± Six s
£¥
.
£×£" $
=£×gj→egTen
=
Itn$**,this is an
d$•mme¥rqy%
.
a,#⇒**±←*%#xK*,¥§g*****Da*=$§,*&¥D[•§g**I***s•$±$§n***⇒EE*amm*d=
: S=IF&*"Pont
.
BadCF)
ItPR##*⇒⇒PoissonSsuummmmationnFformmwloda
.Demand : £ : X - XK ) is continuousThis puts atopaloyy on £
.
Orthogonality Relations for Matrix
Elements of imeps of more generalgroups .
G : Can beany compact group
B$¥ttnF:pEd!d±I!;%eusW@€ : = Set of . irreducible unitaryf. d.reps of G / equivalence€is NOT
A GROUP = IRREP (G).
IN GENERAL.
Consider
: Vu - if �1�- linear tmn
-.
-
.
AE Horn ( Vu ,Vµ ) t.d.clso - Unitary
µ : label distinguishing the imepsofc
""
Imep means p : G → GLCV )
If WCV is a
linear subspace st . pg )WcWfor all W then eigthen
W={o } or we V
Tworep 's are equivflent
if F invertible intertwine
( V, p )NwTsv '
a gifILe 'S ) VgV TV
'
AID T is invertible then
they are equivalent.
For matrix rep's :
Hg pig ) = S'
pg ) 5'
for some Se GLK,
£ )
IRREP (G)
= { All unitary f. d.
cplx rep 's of GKquit .label the distinct imeps
of G by Y
define %EVIE¥HE := { prcg ) A piece
'
) dg ←
Claim : KT is an intertwineie
.
an equivwiant map .
Need toprove
|qq.)E=Ep✓TyFT
ago ) = { Gig ) A pfglg . ) dg
g→g°j = {f(g°g)Apv§5dG*⇐E '
"dgof measure
= { qlgog) Apfg'
) dg
I = qcg . ) A- ✓
E=§*µn*i±t±0¥kI#
The twoinepskaue to be the same .
Ta is a scalar depending on A
To determine it set µ=vand choose an ordered basis In
Yu so the pcg ) become
nmxnm complex matrices.
Set
A=E÷ -
jlijAe = { PG )... pcgibedg
Set k=l and sum :
Tij non = fg p §'
)jkp$e ; dg
* )j ;= Sj
Xij nr= dig.( fadg ) aging- compact
.
WLOG = I
hij
= E -
nr
so
:iBII¥ne¥oms÷feTrese are the orthogonality relations
on matrix elements of imeps .
Remarry Note That we did Notuse unitarily above ! Above is
true forany imed
. cplx matrix repsof Q
But,
for G compact we
can alway change the inner.
product on the group so that,
relative to an ON basis for thatinner product pcg ) is are
unitary Matrices.
( V, p ) any finite did
Q -
rep. of G. ( ginpsotant'
compact )Choose
any nondeg .
inner
product < ly, 4<7 .
define a new inner product :
44, ,4< D :={ < pig )4
, pcg)4Ddg
Still non degenerate .
Unitarily
4 pg )Y, , pg ) YD >
±⇐+, ,4 .
> > any g.
Then of (g) are unitary Wat.
the new inner product . DU
For unitary repsthe orthog .
relations look very beautiful.
¢jeL?@)={v:G→e)f*s9glojd %g- s of Him'S ) ):p
Orthoepy lost is an ON basis
for ECG ) !
fdq%D*otYndg=%a.%
- × -
For G = SUG )
of.
=) Wigner functions
Dominik )
Special cases : sphericalharmonics
,Assoc
. Legendre functions
- × -
In general nice consequence
L4G ) is a G×G aepnn .
¢q.gr ).4)q )
:= 4§iggR )As a G×G rep1 it
ECompletely reducible
KG) I Of Endtf )
= of TroyCorollary G - finite
group .
take dimensions :
lot = Fini+ much much more !
Peter. Weglthemem .
When working with repsit is
often useful to work with
characters :
= (g) : - Ty (pcg ) )basis indpt.
, X✓ ( hghi'
) = Xv (g)
Xµ = Chamfer of I
§Xa*gX✓y)dg=&€
Back to Heisenberg Groups.
Other examples beyond Heissxf )
Exeter : R - Commutative ring
( R = Zn,
Z*I¥r⇐÷HMlaib , c ) MK:b :c
'
)
=
Mante'b±Het£taeb' )
it : M ( a ,b ,a ) → ( ab )
group homomorphism
2 : R -> { M ( as ,c ) }
C -7 M ( 0,0 ,c )
o→R→Heis(RxR)→ROR→o
f ( @,b ) ,@:b'
) ) = ab' @
with Abeliangroups
in
additive notation- -
the cocycle
relation is
f(v , ,rD+f( r.tv . .rs )= f(v, ,k+v3)tf@,V )
Commutation function for �4�
k( ( aid ,@:b'
) ) - ab'
- a 'b
Specialize to R=Z%zrecover the finite Heisenberg
group . Un=1=V"
W=qw qn¥eat÷
¥2 : Clifford algebrasand Extra . special groups .
@
tdbleiden .
smallest matrix reps of this relations.
n=± 8 ,= o'
,Tz= 52
n¥8,
=o'
,k= 02
,03=53
ntl
÷1
05' AEI
III:3]Vq
= £@I Wonkworte
No Bantrcomnutes
WITANn¥ : to ~ -8
,. .
. k,
Yi,
. . .
k,
as above
as in 4×4
8s§=J ]@ I f fnepofel,
v
z 8,
= Nao 'a5 '
¥¥;E¥€¥¥¥D
86=02010-1
These are imed reps of
Clifford .
- Qln= Cliff.
algebra,
y gen's
e,
. - en
The dimension isMls {ei,ejk2d ;
£k]× ZHD
Imep is almost unique .
Vie Matdxdc ) D= 2*3
Vii→ Sri 5
'se Glldic )
anotherrep
-equiv .
Also true that if we scale
Y : →th; tie { ± ' }we also get a rep. of
Qln,
but itmzghtnotteeqiv .
There might be no choice of
S such that
sq .5'
= Ei &
Note that for n =3
n=3 D, Kk = i 12×2
n - 5 8,
- - - B = -It 4×4
Sr,
5'
sk5'
.sB5'
= S (iI⇒5'
aletzxzbutsuppose we change A:→EA ,
with e.ee,
= - 1"
I
FIT = - i 12×2⇒FAt least two distinct imeps .
'
Thing) If n is even
there is a uniqueimed
rep.of Clln of dimension D= 242
( b. ) If n is odd= there are
exactly tug imed reps of
Qln of dimension d=2←⇐k
they are distinguished by the
sign of * valumetorm ":
P,. - - . K
Notice that for nodal
V,
- . . On is central,
ie
Commutes with all the Vi.
we Zan Xs { 0,1 }mod2
Kws=rY-÷jfI¥¥
,
typical element for n= 7. it ,
.
X(011 o 1 01) = Jz 83 050,
:) Kw '
) ⇐ E @
,WY8(w+w#tdf
binary codewords
|£=hmw)|w€zI¥forms a group! !1→zz→E⇒
521 - 0
{ ± I } extension of K"
by Zz.
Cent
Commutatw function
k : Zixzi → { ± ' }
k(w ,w
'
) = ← , )Fg.
wiwj'
When is it non degenerate ?
If we Choose JCW ) ask :
Is there a Kw '
) that anbtommuter
Suppose ?wi=1 mod 2
F io Wio=o then Ji.
antramwto,
¥wi= I mod 2 WE I for
all : ( ⇒ n is odd )then all the ji commute
ondd 8,
.- - K : central .⇒kde"bank
? W ; = o mod 2 F i. wioto
then Jio anticommutes
e.gr#Heanticommo=with Q ) KR , ,k,
Conclusion : n is odd
dos not get a Heisenbergextension because k is degen .
But if n is end we geta Heisenberg extension
.
the..→ZM→|
€ 4 Extra - specialgroup2+2
't ' "
Important point :
.
There is no canonical
presentation of ZE "
as a
product of two maximal
Lagrangian subgroups
I could choose SCZI "
so that ZIE sxfBut there are many differentChoices of S
.
None is
distinguished .
How to give a representationof E ?
You could ch=e an S so
that ain ± sxfand then use SVN rep .
Next themm
: symplectitoni -
see a similar phenomena .
How one the different choicesof reps for different $ 's related ?
Different choices of S are
called fsodarizalions.
"
Like choosing coordinates
and momenta.
Spin GROUP :
-
+ - Kn
ve Pr" denoteV.v :-[ Viv ;
i=l
( do not confuse with
ycw , = Pm ... kw "
we EIZI )Spin Group :
- revenant
Spiny = { t.fm ) . - - - (ten) / ni2=1 }= =
lsisr.
It : Spin ( n ) - 0in )
Ifk ) ) . w=w' defined
byy .w' = - @v)(xw)AvJ
'
ie. Ifk ) ) = Reflection in hyperplae
1- to V