The McKay Numbers

Preliminary Lemmas

4.3 The McKay Numbers

n f & V f . However, by Lem m a 4.2.1 C( g) f is either an. m/ (^-dim ensional general linear group or an m /(p)-dim ensional general unitary group, hence <p/ is uniquely determ ined by a p artitio n p./ of m f ( g ) . We now define the m ap A : F —> P by letting / [if. As 5 2 / ef d/m/(<7) = n, we have

|A (/)| df = n.

/ e P

We let correspond to A. It is easy to check th a t each step of our construction of A is bijective, and therefore we o b tain a 1-1 correspondence between Irr(C?) and the set of maps from F to P w ith desired properties.

L et’s tu rn to th e degrees. We have seen th a t th e p-height of ^ is th e same as th a t of cpg = Yl Vf - By Lem m a 4.2.1, if / € F', then C ( g ) f = GUmf (s )(qdf), so the g-height of <pf is dfn(p'f ). If / € F", th en C{g)f = G L mf (g)(qdf ), so th e ^-height of

<Pf is also dfn(g!f). Therefore, th e ^-height of ip is 5 2 /e F d /n (A (/)').

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Indeed, the irreducible characters of GUn{q) are constructed by Lusztig and Srini- vasan ([LS], also see [FS]). By directly checking w ith th eir construction, we can deduce th e following:

L em m a 4 .2 .3 . I f cp\ £ Itt{ G ) corresponds to the map A as in Proposition J^.2.2, then <p\ lies over p £ Irr(Z {G )) where ap is equal to the product o f the roots of

r w w/)l-

P r o o f o f T h e o r e m 3 .3 .3 . Fix r > 0 and p E lr t(Z (G )). We need to count kr(G,p).

Let rid be the num ber of elements in F of degree d and let fd, i, fd, 2, ■■■, fd,nd be the polynomials in F of degree d. Let A be the set of m aps A : F —>• P satisfying equation (4.1) of Proposition 4.2.2. Given A 6 A, we put

A (/* f) = (4.2)

Also we denote by A' th e m ap from F to P such th a t A '(/) is th e conjugate partition to A( / ) for all / E F . By Proposition 4.2.2, A corresponds to <px E Irr(G ). Set

A(p) = { A E A | <p\ lies over p }.

L e m m a 4 .3 .2 . A E A(p) i f and only i f X' E A(p).

Proof. Recall from section 2.3 th a t for each partition p, |p|

follows from Proposition 4.2.2 and Lemma 4.2.3.

Define hq(A) = hq{(p\) to be th e g-height of ip\.

L e m m a 4 .3 .3 .

fe(G ,p ) = | { A € A(p) | A,(A) = r } | = | { A € A(p) | A,(A') = r } |.

Proof. T he first equality follows from the definition of kr(G, p) an d the second follows from Lemma 4.3.2.

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Now for A E A, we construct a partition p = p(A) of n by defining p = ( jXj), where

xi = dmii = Z) •

d,i d i

= |p '|. Then the lemma

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Observe y is a partition of n because by Proposition 4.2.2,

™ = 53 ^\x(f)\dr = 53d5 3 i = X) ^djmh = 53 53 ^{dmi i =} 53 ^{w -}

fe P d 1=1 d,i,j j d,i j

L em m a 4 .3 .4 . hq( \ ') = n ( y ( X ) ) . Proof. By Proposition 4.2.2,

h*(x') =53 ^{dM x(f)) =} 53 d( E n(x(Ui))= 53 (

²

) dm*i = (4-3)

/ € F d t = l j,d,i ' ^

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Let

U(r) = { y h n \ n ( y) = r }.

For y h n , let

A(y) = { A e A | y (A) = y } and A(y, p) = A(y) fl A(p). By Lenuna 4.3.3 and 4.3.4:

L em m a 4 .3 .5 .

fcr(G,p) = I { A e A(p) I n ( y (A)) = r } |.

Let y = (yXj ) be a partition of n, we next obtain a param eterization of A(p) and A (y, p). Let S{y) be the set of sequences (fi, f2, ■■■, f n) such th a t f j6 F and deg U i) = *j-

L em m a 4 .3 .6 . There is a bijection

s : A(y) -* S ( y) , s(A) = (A , / 2, . . . , / n)

such that A G A(/z, p) i f and only i f

( - l ) na xa%al. . . a" = ap. (*)

where ap ^G is defined in section 3.3 and fo r ¹ ^ j ^ n, aj is the constant term o f fj-

Proof. For A G A(/z) define s(A) = by f j = ITd.ifd,i'x- As € F >

/ j G F . Further

deg {fj) = ^ d m 3di = x jt

d ,i

so indeed s(A) ^G S"^). To see th a t s : A(/z) —»■ SX/z) is a bijection, we define an inverse t : S {p ) —► A(/z) for s. Namely given / = ( / i , /2, - • •, /a ) £ S(m)i / j has a unique factorization

A = IU"?"

d,i

and we define t ( / ) = A G A using equation (4.2). Now

/ <*,»',7 J

so indeed A G A. Further for each j ,

d m d ,i = d e M ) = X 3 d ,i

as / G S(fi), so A G A(/z). By construction, s and t are inverses.

Next by Lemma 4.2.3, A G A(p) if and only if ap is th e product of th e roots of

rw

/ ' A(/)l- But |A(/d,i)| = so

n / |w )i - n / ^ - 1 1= u c ‘d j = £ / / ■

/e P d,t j

F urther th e product of the roots of f j is (—l ) deg^ a j , so A G A(p) if and only if

= U ( - ‘ ^ = ( - D ' l l ^

3 3

as claimed.

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We can now complete th e proof of Theorem 3.3.3. We m ust show kr(G, p) = £

^ew(r) So by Lem m a 4.3.5, it suffices to show

[A(/i, p)| = flp). (4.4)

To establish equation (4.4) we use the param eterization of Lemma 4.3.6, and count th e num ber of / = ( / i , . . . , f n) 6 S(/f) satisfying (*). L et

f ( x ) = x m + + . . . b i x + bo E F,a[x].

For 1 ^ i ^ (m — l) /2 , there are q2 choices for 6t-, and th en by Lem ma 4.3.1, / G F if and only if bo G 6m_» = bob*, and if m is even, b^*2 = bo. Thus there axe g2(deg(/,)—1 )/2 _ gXj- 1 choices for the coefficients bjti, . . . , 1, and the coefficients o,j = bjfi m ust satisfy (*). By Lemma 2.3.3, there axe 0 ( n , a p) tuples (dj.,...,O n) satisfying (*), so

3

as desired. This completes th e proof of Theorem 3.3.3.

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We now prove the following proposition which is equivalent to Proposition 3.3.6.1.

Recall n = 2m or 2m -+-1 and I = [m].

P r o p o s it i o n 4 .3 .7 . Let G = G{7n(Fg). 27ien

E . / c / ( - 1 ) l'/|f c d ( A / , p ) = E '

where ^ is the same as in Lem m a 3.3.1.

Proof. Use induction on m . If m = 0, th en n = 1, so G = GUi(q) = Z( G) =

As I = [0] = 0 and Hq+i has q + 1 irreducible representations, all of degree 1, if follows th a t

^ 1, i£d = 0;

E c - i m w ) = w , p ) = ^

J'cy 0, otherwise.

O n th e o ther hand, 1 has a unique p artitio n p = (1), with n{p) = 0 and P(p, ap) = 1. So the right-hand side of the proposition is 1 if d = 0 and 0 otherwise. Therefore, the proposition holds for m = 0.

Assume m ^ I. Let A = A(L). For 0 ^ I ^ m, let A j be th e set of J C / whose minimal m em ber m in(J’) is equal to L Here we set min(0) = 0. T h en A is the disjoint union of Ai, 0 ^ I ^ m.

If J E Aj and I > 1, then L j = G+l x L”,'2* where G+z = GL[(q2) and L",-2/ is the Levi subgroup of GUn-7i(q) corresponding to the subset J ' = { j — 111 < j E J}.

Let Z \ be th e subgroup of Z (G +l) of order q + 1 and Z i = Z(G U n-n (q )). Then Z( G) < Z \ x Zi . So by Lemma 2.2.5 we have

d

J = 0 Pi&Zi P iP 2 = P

Let Z = Z (G +f). Fix p E I n r ^ ) . <p E Irr(G +i) lies over p if and only if <p lies over some r E Irr(Z ) w ith r lying over p. So Irr(G +i, p) is th e disjoint union of Irr(G +i, r ) for r £ I r r ( ^ ) lying over p. As Z = 0 , 2 there are (g — 1) choices of r E Irr(Z )

lying over p. In this case r lies over p if and only if the restriction o f r to Z \ is p. So

a p = P { z q~ ' ) = r ( z ^ x ) = = o T l .

Therefore, by Lemma 3.3.1,

kd{G*L,p) = J2 WG+', ^t )= Y, E ,»)

relrr(S) 6€ET(J2 _ 1 PiH

t I z i = P b " ~ 1= a p n ( p i ) = d /2

MlH

n { ji \) = d /2

Therefore, by induction

^(-ir'kd(Lj,p)=~Y E >=AG +t,Pi)Y(-l)'J''k*-i(-L-’''i> *)

J e A t j= o P1.P2 / ' c / '

P1P2—P ~

= - E E r( E

J = 0 P1.P2 p 2H i - 2 l P1P2 P n ( p 2)= d —j

for all 1 ^ I ^ m , where

pi H n(pi)=j/2

Recall from section 2.3 th a t if p i fr I w ith n(px) = j / 2 an d p2 I- ti — 21 with

71(^2) = d —j , th en p = 2 p iU p 2 1“ n w ith n (p ) = 2n(pi) +71(^2) = d. So when j runs over all possibilities, p and (p 1} P2) run over ail p 1- n w ith n (p ) = d, and such th at p = 2pi U P2 for some p i I- 1 and P2 H n — 2Z. Then let Z ru n over all possibilities from 1 to m , p and (p i, P2) ru n over all p I- n w ith n(p) = d such th a t p can be w ritten as 2 p i U P2, except th a t p i 7^ (0), as 1 ^ |p i| < m. Therefore, by exchanging the order

of sum m ation,

E (-1

)'J'kd( L j , p )

= E E

( - 1 ) 'JI kd( Lj , p )

q^j c i i=i Je&i

= - 53 X X) 92Cf(/il)~<y(m ))^i,api)^(^2,aP2)-

(m i,M2) 0P.)g l0

n ( ji) = d j* =2miUM2 P P Im iI #o

On th e o th er hand, we need to show

/

^ ( - l ) |J|fcd(Lj,p)

= '

2 2 0

(p, a„)=

] T 0

{fJ.,ap)

J C .I fi h n

n ( ji) = d

and ^ ap) can be regarded as th e term corresponding to 1/zjJ = 0 in the sum a, where

<r- E E E ^{42 <' 0} “)_<('‘'))i 3 (/ii,apl)/J(M 2 .a«)-

So as G = Lq, it suffices to show th a t a = kd( G , p ) = Y .

fil-n n (p i)= d

where ijl\ is allowed to be 0. However, for each fj. h n, by Lemma 2.3.7 and Lemma 2.3.8,

X^ X 92(i0il)_<yail))^(Aii»aPi)/3(^2,aP2) =

(M i .M2) nPx)gl0 (£=2/11 U/i2

^ 9 2 ( / (m i) - 5 (m i) ) ( ? _ 1 ) ^ 0 ( 9 + a p ) = 0 p ) .

(m i .M2) M = 2m iUM2

Therefore, th e proposition is proved.

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