Physics 618

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

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.

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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

[email protected]

: 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

Now for me = 2m even.

There is a ! imep of

Qln

But two in equivalent imeps

A-± of Spin Cn )

p (V.v, ) . . - A.k)P± ±

¢even )

P±= k(±K,- - - . kn )

5 chosen so that

Cs8 ,- - .KIH +11

.

P± proj .

to Yz dime

dime At = 2mi '

= 2¥ ' . '

.