PARTITIONS, IRREDUCIBLE CHARACTERS, AND INEQUALITIES

TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 325, Number 2, June 1991

PARTITIONS, IRREDUCIBLE CHARACTERS, AND INEQUALITIESFOR GENERALIZED MATRIX FUNCTIONS

THOMAS H. PATE

Abstract. Given a partition a = {ax , a2, ... , as} , ax > a2 > ■ ■ ■ > as, of

n we let X denote the derived irreducible character of S„ , and we associate

with a a derived partition

a = {a, — 1, a2 — 1, ..• , ott — 1 , at+x , ... , as , 1 }

where t denotes the smallest positive integer such that at > a(+1 (aJ+r =

0). We show that if y is a decomposable C-valued n-linear function on

Cm x Cm x ■ • ■ x Cm («-copies) then (Xa Y, Y) > (Xa, Y, Y) . Translating into

the notation of matrix theory we obtain an inequality involving the generalized

matrix functions dx and dx ( , namely that

(Xa(e))-xdx(B)>(Xa,(e))-ldxAB)

for each n x n positive semidefinite Hermitian matrix B . This result gener-

alizes a classical result of I. Schur and includes many other known inequalities

as special cases.

1. Introduction

If c £ CSn , the group algebra obtained from C and the symmetric group on

{1,2,... ,«}, then we define the generalized matrix function dc by

(1.1) dc(B)=52c(o)f[bia(i)o€Sn (=1

for each « x « matrix B = [ft..]. If c(e) / 0 then by dc we mean (c(e))~xdc.

Of particular interest are the immanents, the generalized matrix functions dx

where X is an irreducible character of Sn. Familiar examples are det(-), the

determinant function, obtained by setting c = s, the Signum function, and

per(-), the permanent function, obtained by setting c(o) = 1 for each o £ Sn .

There are many known inequalities that involve restricting the functions dc

to %fn , the « x « positive semidefinite Hermitian matrices. Perhaps the oldest

Received by the editors May 31, 1989.

1980 Mathematics Subject Classification (1985 Revision). Primary 15A15; Secondary 15A69,15A45.

Key words and phrases. Generalized matrix function, tensor product, induced character, part-

ition.

©1991 American Mathematical Society

0002-9947/91 $1.00+ $.25 per page

875

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

876 T. H. PATE

is the classical Fischer inequality, which states that if B £ %* is partitioned

in the formBXX BX2

.B2X 522

where Bxx is « x « and B22 is p x p then

( 1.2) det(fi) < det(Bx, ) det(2?22)

with equality if and only if B has a row of zeroes or Bx2 is the zero matrix.

In 1908 Schur, see [3], proved that if X is a character of Sn then

(1.3) det(B)<dx(B)

for each B £%?n. For a short proof see [4]. Hence, the determinant function is,

in the sense of (1.3), the smallest of the normalized immanents. This naturally

led to speculation as to which, if any, of the normalized immanents might be

largest, in the sense of (1.3).

In the sixties M. Marcus proved, see [5], a partial analogue to (1.2) involving

permanents, namely that if B = [ft. ] £ %fn then

(1.4) per(5)>ftllPer(5(l|l))

where B(s\i), I < s, t < «, denotes the (« - 1) x (« - 1) matrix obtained

from B by deleting the 5th row and rth column.

Moreover, Marcus conjectured that if B £ ^n+p is partitioned as in our

statement of the Fischer inequality then

(1.5) per(ß)>per(5,,)per(Ä22).

This result was later proved by Lieb, see [6], and subsequently generalized to

the symmetric algebra by Neuberger, see [1].

The similarity between (1.2) and (1.5) led naturally to the conjecture that

(1.6) dx(B)<per(B)

for each B £ %An ; so the permanent function is believed to be the "largest" of

the normalized immanents. Despite considerable effort (1.6) is still unresolved.

Efforts to prove (1.6) have nevertheless led to the discovery of some interesting

theorems, some of which provide information not implied by (1.6). James

and Liebeck, see [8], proved that if X is an irreducible character of Sn and its

associated partition is of the form {p, q, lr}, p+q+r = « , then (1.6) holds for

each B £ %An . The author, see [9], proved that dc(B) < per(B) for all B £ßivn

and all c £ £Sn for which there exists an / £ CSn and aAc{l,2,...,«}

such that

(1) c(o) = Y,J(ox)f(T) each o e Sn,(2) xf = f for each r £ Sn such that t(A) = A.

It is easy to show, see [9], that all irreducible characters of Sn derived from

two-term partitions are expressible as sums of such functions c. Hence, this

theorem includes the r = 0 part of the James-Liebeck result as a special case.


GENERALIZED MATRIX FUNCTIONS 877

Heyfron, a student of James, see [10], considered the single-hook characters,

characters derived from partitions of the form {q, ln~9} , and proved that the

associated normalized immanents increase with q. In other words, if A', is

derived from {q, ln~9} and X2 is derived from {q+1, l"_<?_1} then

(1.7) dx¡(B)<dXi(B)

for each B £%'n. This result was originally conjectured by Merris and Watkins,

see [11], who proved (1.7) in case q = 0, 1, or « - 1. It will be shown that

our main result generalizes Schur's inequality by replacing (1.3) with a chain of

inequalities starting with dx(B) and terminating with det(2?). Moreover, our

main result reduces Heyfron's result to a special case and, in conjunction with

[9], implies and extends the James-Liebeck result.

2. NOTATION AND BACKGROUND

We let V denote Cm for a fixed positive integer m which will usually be

clear from context. For « > 1 we let Tn(V), often abbreviated to Tn , denote

the set of all «-linear complex valued functions on V. If « = 0 then Tn(V)

denotes C. Given an inner product ( , ) on V we extend ( , ) to each of

the spaces Tn by choosing an orthonormal basis {ei}m=x for V and defining

mm m

(A>B) = EE-E^,> v ••• '\)BK' v- '\]9, = 1«2=1 Q„ = X

for each A, B £ Tn. Note that this extended ( , ) is independent of the

orthonormal basis {e,}™ j •

If A £ Tn and B e Tp then the tensor product of A and B, denoted by

A ® B , is the member of T such that if xx , x2, ... , x £ V then

(A®B)(xx,x2, ... ,xn+p)

= A(xx, x2, ... , xn)B(xn+x, xn+2, ... , xn+p).

Note that \\A ® B\\ = \\A\\ ■ \\B\\.We now define an action of Sn on Tn which we extend to CSn in such a

way that Tn becomes a left CSn module. If o £ Sn and A £ Tn then by o A

we mean the member of Tn defined by

(oA)(xx, x2, ... , xn) = A(xa^ , xa{2), ... , x ff(n))

for xx, x2, ... , xn £ V . Clearly, (ctt)^ = ct(t^) for o , t £ Sn , and eA = A

where e denotes the identity. Since each xp £ <CSn is represented in the form

J2oes V(a)a we define y/A to be J2a y/(o)oA. Note that if A, B £ Tn

then (oA, B) = (A, o~xB) for o £ Sn and, consequently, if y/ £ CSn then

(y/A, B) = (A, y/*B) where y/*(o) = y/(o~x) for each cr e Sn.

A member of A of Tn is said to be decomposable if there exists /, , f2, ... ,

fn £ V*, the dual of V, such that A = fx <g> f2 ® ■ • • ® / . In this case


878 T. H. PATE

Mil = lXi ll/J and if o £ Sn then o A = fg-l{x) ̂•••®fa-x{n) so, if xx,x2, ... , xn are members of V such that ft(y) = (y, x¡) each y £ V and

1 < / < « , then cM = (jcff-i,j,)* ® {xa-x,2A* ® • • • ® (•*<,-'(„))* where the conju-

gate linear map z —> z* is defined by z*(y) = (y, z) for v , z g F . Note that

(x, y) = (y*, z*). Consequently,n n

(x\ ® x\ ® • • • ® x* , v* ® y* ® • • • ® y*) = JJ(x*, y*) = JJty,., *,.);=1 ;=1

for xt, yt£ V, 1 < i <n.Inequalities involving immanents translate simply into inequalities involving

decomposable tensors. Moreover, the converse is also true. The connection

between these two types of inequalities is presented in the following:

Lemma 1. If y/{, y/2£ CSn then dv (B) < dv (B) for each B £%?n if and only

if (y/xY, Y) < (y/2Y, Y) for each decomposable Y £ Tn.

Proof. If B £ J^ then there exist y, , y2, ... ,yn £ V(m > n) such that

ft; = (y., y;), 1 < i, j < n . Letting Y denote y*x®y*2®- • • ® y* we have, for

xp £ CSn,

dv(B) = £ y,(o) n bia(l) = ¿2 vio) f[{ya(l), y;> = E vW U(y,. V(o>rr ;'=1 (T 1=1 O" |"

= E ^(ct)^-'(d ® • ■ • ® ̂ -'w » ̂ î ® • • • ® y*n)a

= Y,^{o)(oY,Y) = (xpY,Y).a

Hence, if (y/xY, Y) < (y/2Y, Y) for each decomposable Y £ Tn then dv (B) <

d (B) for each B £ Tn. The proof of the converse is similar, and omitted. D

When dealing with the characters of the symmetric group our notation follows

Marcus, see [7,. Chapter 6]. If a = {ax, a2, ... , at} is a partition of « , we

always assume a, > a2 > ■ ■ ■ > a(, and tp £ Sn then by Da we mean the

Young diagram whose first row contains tp(l), tp(2), ... , tp(ax), whose second

row contains tp(ax + l), tp(ax + 2), ... , tp(ax + a2) etc. By Ra(p and Ca(p

we mean, respectively, the row and column groups of Da , and by ra and

ca we mean the corresponding row and column symmetrizers. Explicitly,

r ,„ = Y]ct where summation is over R ,„, and c„ _ = Y,£(o)o where the

summation is over Ca , so it is perhaps more accurate to refer to ca as a

column skew-symmetrizer. The Young symmetrizer I? , associated with a

and c/?, is then ra ca . We denote the irreducible character associated with

a by Xa .

We require a formula for Xa in terms of the associated Young symmetrizers.

According to [12, Theorem 1, p. 108], we have

(2-1) w = 5f-E*..f(*",ff~'*>



where na is the dimension of the representation which is simply Xa(e). For

/ e CSn we let / = (f(e)yxf, f(e) ¿ 0, so, since na = Xa(e), we have

(2.2) Xa(o) = (n\)-x^gac(g-xo-xg)

g

but X and X are real-valued class functions so X (o~x) = X (a) and

(2.3) Xa(o) = («!)-' Y.K,^S~Xog)g

which implies that

(2.4) X =(n\)~XYor c o~Xa

for each tp £ Sn. The multiplication oxa in (2.4) involves a slight abuse of

language since o £ Sn and ra £ CSn . In such cases we identify o with the

member of CSn that assumes the value 1 at o and 0 elsewhere.

As is clear from (2.1) the character Xa is independent of tp and, conse-

quently, tp may be chosen according to convenience. Since tp will always be

clear from context we delete tp from the notation and abbreviate as follows:

R = R , C =C , r =r , c =c , % =% ,and D =D .a & ,<p ot a ,<P ot oi ,(p 7 a a ,(pJ a a ,(p ' a a ,<p

Lemma 2. If y/, % £ CSn and IT2 = kïï for some kÔ then

(2.5) ^rj^ff"1 = ô%yio~x = k~xâ%yi%o~x.

a a a

Proof. If f £ CSn and x £ Sn then, by setting u = xo~x, one sees that

ôfxo~x = Yû~Xxfu = ôxfo~X,a u a

an equality that immediately implies that if /, g £ CSn then

ôfgo~x =^2ogfo~x.ct a

Consequently, we have

Y,oy/%o~x = k~xôyi%2o~x = k~l ôgy/go'1

a a ct

as required. D

Combining Lemma 2 with formula (2.4) we see that

(2.6) X=(n\\R A)~XYor c r o~xa

and

(2.7) X=(n\\C I)"1 Yoc r c o~XCT

for each tp e Sn.


880 T. H. PATE

If G is a finite group, and X is a character of H then X' , the character

of G induced from X, is defined by

(2.8) XT(M) = |//f1E^(cT-1Wc7)

where summation is over all o £ G such that o~xuo £ H. See [13, p. 30], for

details. Extending X £ <CH to all of G by defining X(o) = 0 for o £G- H

we see that (2.8) immediately implies that

(2.9) XT = \H\~XJ2°Xa~l

and, since X](e) = X(e)\G\/\H\,

(2.10) P = (X(e)\G\rXJ2°x°~l-ct6G

If A c {1, 2, ... , «} then by G(A) we mean the set of all o £ Sn such

that er(A) = A and o(i) = i if i ^ A. The idempotents A? (A) and s/(A)

are defined to be |G(A)|_1 lZC€G(A)(7 anc* |G(A)|_1 Z)ct6G(A)e((7)(J respectively.

These idempotents provide necessary factorizations of the symmetrizers ra and

c , for if the row sets of D are A, , A,, ... , Ar and the column sets are T, ,a ' a 1 ' I ' ' 5 1'

T2, ... , Tr then

(2.11) rQ = n|G(A,)|^(A,.);=1

and

(2.12) ca = Y[\G(T])\^(Yj).7=1

All subsets A c {1,2,... , N} with |A| = n give rise to groups C7(A)

isomorphic to Sn . For our computations we require a definite isomorphism

o —y oA from G(A) to Sn , so we let ô be the unique increasing function from

{1,2,... , «} onto A and define o = ô~ öS for o £ G(A). Given two

disjoint subsets A, and A2 of {1,2,... , N} with |A[| = « and ¡A2| =p we

require two isomorphisms, " A " and " V ," each as described above. Letting H

denote G(AX) ■ G(A2) the map y —» (yA , yv) is then an isomorphism between

H and Sn®Sp.To define CSn ® CS we consider the special case A, = {1,2,... ,«} and

A2 = {n + 1, n + 2, ... , n + p} and require that cr ® x, o £ Sn and i£Sp,

denote the member of Cr7 such that (cj®t)(0) = 1 or 0 depending on whether

o = 6A and x = 0V , or not. Then, we extend ® to the rest of CSn x CSp by

requiring

/®S= E E /((T)^(T)tJ ® T-ct65„ tes„



In this way CSn ® CS is identified with a subalgebra of CSn+p , namely CH.

There are certain simple properties satisfied by " A ", " V ", and ® which we

present without proof:

Lemma 3. Suppose Ax = {1,2,... ,«}, A2 = {n + 1, n + 2, ... , n + p},

H = G(AX) ■ G(A2), o£H, f£CSn, geCSp, A £ T„, and B £Tp. Then,

(1) o(f®g) = ohf®oyg,

(2) (f ® g)o = fo* ® go" ,(3) o(A®B) = (ohA)®(oyB),

(4) (f®g)(A®B) = (fA)®(gB).

Returning now to the context of formulas (2.8), (2.9), and (2.10) we see that

the map I-»l' need not be restricted to characters X of H. Indeed, we

may define f by formula (2.8) or (2.9) for any / e CH. Consequently, we

state the following:

Lemma 4. Suppose f £ CSn and g £ CSp are class functions such that f(e) =

g(e) = 1. Then,

(f^g)' = (n+„pYlE<p(f®s)<p-1v y <peu

where U denotes the set of all members tp of Sn+p such that tp restricted to

{1,2,... , «} is increasing and tp restricted to {n + 1, n + 2, ... , n + p} is

increasing.

Proof. Let H = G({ 1, 2,...,«}) • G({n + l,n + 2, ... ,« +p}) and note that

U is a set of distinct representatives of the left cosets of H in Sn+ . The rest

of the proof is now a straightforward computation:

(«+p)!(/V^)t = ô(f®g)o~x = EE^T^®£)T~V'ct re// ¡peu

2_^<?(X f(x ) ®T g(x ) )tpz€H<p€U

= \H\Y,<Pif®g)<P~X><i>etj

where the penultimate expression is obtained from its predecessor using Lemma

3. Since \H\ = «!p!, the proof is complete. G

3. Main results

If / £ CSn then we shall write / > 0 if (fY, Y) > 0 for each decomposable

Y £Tn. Hence, f < g if (f Y, Y) < (gY, Y) for each decomposable Y £ Tn .Our main results are Theorems 3 and 6. Theorem 3 provides a stepping-up

inequality: it shows how, given an irreducible character y/ of Sn, to find a

second character f , induced from a Young subgroup of Sn , such that y/ < §? .


882 T. H. PATE

In the same sense Theorem 6 provides a stepping down inequality. We combine

Theorems 3 and 6 to obtain Theorem 7, which is the result mentioned in the

abstract.

If k > 1 then A (V) denotes the set of all alternating A:-linear functions

on F,and, if A£ A"(V) and B £ AP(V), then A A B denotes sf(A)(A ® B)where A = {1, 2, ... , n+p} . The following appears in [14].

Theorem 1. If A £ An(V), B £ AP(B), and either A or B is decomposable

then

\\AABt<(nXP)~X\\AU\B\t

In the context of Theorem 1 decomposability means decomposability with

respect to the multiplication in the exterior algebra, so the statement that A e

A"(V) is decomposable means that there exists xx, x2, ... , xn £ V such that

A = x* A x2 A • • • A x*n . Our purposes require a strengthening of Theorem 1.

Theorem 2. If A is a decomposable member of Tn, A, = {1, 2, ... , «} , A2 c

{« + 1, « + 2, ... , n + p + r} with |A2| = p, and A denotes A, u A2 then

\\sf(A)(A®B)\\2<(n+nP^ ||(j/(A,))A^||2||(j/(A2))vfi||2

for each B £ Tp+r.

Proof. First, note that it is permissible to assume A2 = {«+l,«-|-2,... , n+p} .

Let {ei)7=x be an orthonormal basis for V . Then,

||j/(A)(^®5)||2= E WmA®B)(eq)\2

= E E \*mA®B)'eq,et)t<i^„+p mterr

where:

(a) for positive integers s, t, Ts t denotes all sequences of length 5 each

of whose terms is a member of {1, 2,...,/},

(b) stf(A)(A ® B)(en) denotes s/(A)(A ® B)(ea , e. ,... , ea ) for q £

r„+/7+fim,and

(c) sf(A)(A®B)(e,et) denotes

s/(A)(A ® B)(e ,e ... ,e ,e e ... ,et)"\ ^2 "n+p I 2 r

for q£Tn+p m and t£Tr m.

Since j/ (A) does not effect the last r places of B we let Bt, t£Trm, denote

the member of Tp such that if z, , z2, ... , zp are in V then

Bt(zx ,z2,...,zp) = B(zx, z2, ... ,zp,eh,eh,... ,et)

and note that

s/(A)(A ® B)(e , e,) =sí(A)(A ® B,)(e)



for each q £ Tn+pm . Hence,

ii2\W(A)(A®B)\\2= E E \^(^)(A®Bt)(eq)\2

'err,mî€r„+p m

= E IIÂ)^®^)!!2.terr

Now, by Theorem 1,

K(A)(^(A1)A^®^(A2)v5i)||2<("+P) \W(Ax)AAfW(A2)yB\t

But, j/(A)er = (T^(A) = e(ff)j/(A) for each cr 6 C7(A). Hence, sf(A)s/(Ai)

j/(A), i= 1, 2, and

j/(A)(j/(A,)Ayl®j/(A2)vJf?/)=j/(A)j/(A1)^(A2)(^®5i)

= ^(A)(^®5;).

Therefore

||^(A)(.4®5)||2= E K(A)U®fit)||2i€T,

< n+P\ X~* WrWIK \^ A\\2\\r^f\ \VDll2

« E K(A,)A,4||V(A2)V5'er,

+ *) HÂ,)^!!2. E IK(A2)v5r|« +P

'6rr

"+P) |K(A^||V(A2)V¿?||2. D

Given y = y* ® y2 ® ■ • ■ ® y^ where y, , y2 , ... .yê F and n < N we

shall need to express Y as T, n ® T2 n where T, n = y* ® y2 ® • • • ® y*N_n

and Y2 n= y*N_n+x ® y^_n+2 ® ■ ■ ■ ® y# • Since « will usually be clear from

context we write T, instead of 7, n, and T2 instead of T2 n. Moreover, if

cr £ Sn+p then Yx(o) denotes y*(1) ® y*(2) ® • •• ® y*(A,_„} and 72(cr) denotes

yCT(AT_„+i) ® yCT(Ar_„+2) ® • ■ • ®y*o(N) ■ Given tnese definitions it is easy to see that

if o £ Sn+p then o~xY = Yx(o) ® T2(cr), and if t € H = G(AX) ■ G(A2), where

A, = {1, 2, ... , N-n} and A2 = {N - n +I, N - n + 2, ... , N}, then

(cTT)-1y = T-1(cT-1T)

= ((rA)-1®(rv)-1)(F1(c7)®y2(c7))

= (t-')Ay1(c7)®(T-1)vy2(cT).

Theorem 3. If a = {ax, a2, ... an+p} = {a, , a2, ... , a , l"} is a partition

of N, ß = {ßx, ß2, ... , ß}, it is associated (or transpose) partition, and a


884 T. H. PATE

denotes {ax, a2, ... , ap} then

(XY, Y) < (N\ E^'W' Yx(tp))(X{nY2(tp), Y2(tp))^ ' </>€U

for each decomposable Y e TN, where U denotes the set of all members of

SN that increase on {1,2,... , N - «} and {N-n + l,N-n + 2,... , N}.

Consequently,

xa<(xa,®x{n)\

Proof. By (2.4) we have

X = (N\)~x Y or c o~X

o€SN

for any tp £ SN. In the following the underlying Young diagram is the nat-

ural Young diagram obtained by setting tp = e. Moreover, we abbreviate ra

with r and ca with c. We let c denote (n+p)\sf(Ax) and c" denote

Y['j=2\G(Aj)\s/(Aj) where A,, A2, ... , A, are the column sets of Da. Note

that c is the symmetrizer associated with the first column of Da and c" is

the product of the symmetrizers associated with the other columns. Hence,

c = c'c" = c"c . Now, by (2.6), we have

N\X =\R \~XY°rcro~X = \R \~X Y ore c ro~Xa o

= (n+p)$\Ra\\Ca$-xJ2°rc'(c")2ro-xa

= (n+p)$\RJ\Ca$-Xôrc"c'c"re-XCT

since (n+p)$c"f = \CJc" . Let K = (n + p)\(\RJ\Ca\N$-x . Now,

(XJ, Y) = KY,(orc"c'c"ro~XY, Y)

= Kj2(c'(c"ro'XY),c"ro-XY)CT

= Kj2(c'(c"r(Yx(o) ® Y2(o))),c"r(Yx(o) ® Y2(o)))a

= KYá{c\{c"rfYx(a) ® Y2(o)), (c"r)AYx(o) ® Y2(o))a

= (n+p)\Kj2\\^(\)((c"r)AYx(o)®Y2(o))\\2a

<(n+p)$n+nP\ K^\\jt/(A')((c"r)AYx(o))\\2\\sf(A0)Y2(o)\\2^ ' CT

by Theorem 2, where A' = A, - {N - « + 1, N - « + 2, ... , N} and A0 =

{1,2,... ,«}. Letting % be as in the statement of the theorem and observing



that s/(A')c" = (p$~xcal and ra = ra, the above is

= n\p\Kj2\$Px-rlca'rarYx(o)\\2\\s/(A0)Y2(a)\\2a

= n\(p$~XK E E IK^W)II V(A0)T2(ç*t)||2

= «!(p!)-lÊEiK'ra'(TArlyi^)ii2nj/(Ao)(TV)"ly2(9')ii2

= n$p\yXKj2 E T,K''a'«~lYi(v)ñ*($0~lY2(9)\\29 ueSN_n6€Sn

,-l„r^v^, -!«•/ n -!•«•/ un.«. v/.-li= n!(p!)-ÊE<ctt'ra'M *»' c*'ra'u Yx(tp))\\^(AQ)e-xY2(tp)\\¿

</> u,e

= n$p$-X\Ca,\KY,Y.ûra'ca'rau~^Yx((P)^Yxi(P))ip u,e

x(dtf(A0)6-XY2(<p),Y2(tp))

But dtf(A0)d~x =sf(A0) for each d £ Sn and

E^'CQ'ra^_1 = lâ'lEâ'^'^"1ft ß

by Lemma 2. Since \Ra,\ = \Ra\, \CJ= p\\Ca$(n+p)$~x, and E^P^-c^/T1

= (N - n)\Xa, by (2.4), the above is

= n\\Ra\\Ca$N-n)\K((n+p)$-xYá{XaiYx(tp), Yx(tp))(X{nY2(tp), Y2(tp))

= (Nn) Eâ'W> W^^W' W>:

which, finally, is the expression appearing in the statement of the theorem. That

Xa < (Ç®1{1»})T

now follows from Lemma 4. D

We now present the notation necessary to translate Theorem 3 into the lan-

guage of matrix theory.

If 5 < « then Qs n will denote the set of all strictly increasing sequences of

length 5 each of whose terms is a member of {1, 2, ... , «}. If tp £ Qs then

the sequence complementary to tp , denoted by tpc, is the member of Qn_s n

whose range is complementary to the range of tp. If C = [c ] is an s x t

matrix, p £ Qs s, 6 £ Qt , then C[p|t9] denotes the s0 x tQ matrix whose

/7th term is c^,¡, e{j). Similarly, C(p\tp) denotes the (s - s0) x (/ - tQ) matrix

whose /7th term is c e,¡, ec,...


886 T. H. PATE

Now, if C = [c¡j] = [((y;., y,)] £ %?N, then by Theorem 3 and the proof of

Lemma 1 we have

dx(C) = (XY,Y)

AT"1I n E<*a'W> Yx(tp))(X{nY2(tp), Y2(tp)).

But, tp £% restricted to {1,2,... , N - «} is a member of QN_n N and

(X^x„^Y2(tp), Y2(tp)) = det(C(tp\tp)). Hence, we have the following:

Theorem 4. If C = [ci}] £ ß?N, a = {ax, a2, ... , a , l") is a partition of N,

and a = {a, , a2, ... , a } then

^,,(C)-(«) S dXi(C[tp\tp])det(C(tp\tp))9£QN-n.N

with equality if C is diagonal or both sides reduce to 0.

Lemma 5. Suppose A £ Tn, B £ Tp, and y £ Sn+p satisfies:

(a) y2 = e.

(b) If 1 < i < n and y(i) ^ /' then y(i) > n .(c) Ifn + l<i<n+p and y(i) ^ /' then y(i) < n .

Then (y(A ® B), A ® B) > 0.

Proof. Conditions (a), (b) and (c) imply that y is a product of disjoint trans-

positions (rs) such that 1 < r < n and «+l<5<«+p. We claim

that we may assume y = (1, n + 1)(2, n + 2) ■ ■ ■ (k, n + k) where k is the

number of transpositions involved in y, for if this is not the case then there

exists a £ G({1,2, ... , «}) and ß = G({n + 1, n + 2, ... ,« +p}) such that

y = (aß)~x(l, n + 1)(2, n + 2)---(k, n + k)(aß) ; then

(y(A ®B),A®B) = (y'(aAA ® ßvB), aA A ® ßvB)

where y denotes (1 , « + 1)(2, « + 2) • • • (k , « + k). Now, assuming y is

(l,n+l)(2,n + 2)-..(k,n + k),we have (y(A®B), A®B)m m

= E"" E 7(A9B)(eqt , eq2,...eqJ(A®B)(eq^, V ... , eqJ

= Eî ,'••• '*« >>*«, ,'■•■ '*« )BK>"- >\>eq , .'••■e, )

x^fi,...,* e . ,e )

x5Kn+,'--'^'ew+1'V.+,'-"'^V

which, if for each 5 , /, Ts f denotes the set of all sequences of length 5 each

of whose terms is a member of {1, 2, ... , t}, is the same as

E E E E AC . 0*('fl, et)A(eq , er)B(es, et)q r s t



where the summations are over Tk m, Tn_k m , Tkm and Yp_km respectively,

and A(e., e) denotes A(e. ,e, ,... ,e, ,e. ,e. ,... ,er ), etc. The lasts r' sx s2 sk rt r2 rn-k

expression above is the same as

= E YdA(es,er)B(es,et) >0. n

Lemma 6. Let A= {Sx, S2, ... , ôn+p} C {1, 2, ... , N + P}, A; = A n {1, 2,

... , N}, Ar = An{N+l, N + 2, ... , N + P}, and k = min{«, p} . Suppose

« = |A | and p = |A |. Then

-1 k

A7(A)=(n+nPy E(") (¿)^')^(A')^(Ar)y(A')

where y. = \Xs=x(ôs » <*„+,) > 1 < i < k, and y0 = e.

Proof. Let H denote G(Al)-G(Ar). It can be shown, see [2], that {y0, yx, ... ,

yk} is a system of distinct representatives of the double cosets of H in G(A).

Moreover, if a list is constructed of all products of the form oytx, o, x £ H,

then each member of Hy¡H will appear in the list [(« - i)\(p - /')!(/'!)] times.

Letting ci denote the reciprocal of [(« - /')!(p - /)!(/!) ] we have:

(«+p)!^(a)= e * = í>< E ^ = X>/(e*U(£t)ct6G(A) i=0 CT.retf ;=0 \a€H J \x£H J

k

= |//|2 E^(A!)^(Ar)y/^(Ar)^(A/)

i=0

= n\p\ E ( 1 ) (P. ) A^(A,)A?'(Ar)yiA^(Ar)^(Al).

Therefore,

A7(A)« +p

n

[=0

-1 k¿("j (Pi)^(Al)A^(Ar)y¡S-(Ar)^(Al). D

Theorem 5. Suppose A,, A2, ... , Ak are pairwise disjoint subsets of {1, 2,

... , N + P}. For 1 < / <k let A¡=A,.n{l, 2, ... , N} and A' = A(xx{N+ I,

N + 2,... , N + P}. Denote |A(.| fty«,, |A'| by p¡, and ni+pl by m¡. Then

nC;)i=i x ' '

\\a?(a¿(a®b$=1

> n^(4)\i=X

f k \v

n^K) b\i=X J


T. H. PATE

for each A £ TN and B e Tp. Equivalently,

U[^)(jl^{ài)(A®B),A®B\

Proof. Let A¿ = {Sn,S¡2,... ,Sjm} where we assume SiX < Si2 < ••• <

àim., 1 < i < k. For 1 < j < c¡, c; = minf«;, p(.}, we let yi} denote

ri/=i(^M ' a¡ s+n), a product of disjoint transpositions. Now, by Lemma 6,

n(:0iVw=n it ( nj ){pj) ^(a;)^(a>,^(a;)^(a;)

- E E ■ • • E ft ( /' ) iPi ) ̂ m^yÂ])^).j.=oj2=o a=0/=i Wi/ V7'/

Given A £ TN and B £ Tp we have

k

n^(A;.)^(A>u. ^(a;)^(a;.) j (¿ ® b) , a ® 5

■((S^)(ffiHA"(5Hv')"

=/[n^)^'®5')^'®Ä')

where Ä = (Y[U^(A$)aA and B' = (YlUî^)) B ■ But [&, fy » aproduct of disjoint transpositions (rs) such that 1 < r < N and N + I < s <

N + P . Therefore,

((n^,)(^®5')^'®5')>°



by Lemma 5. Therefore, by setting each j. = 0 we obtain the inequality

k

>

j5*(Ai)(A®B),A®B)

\&(Al^(A\)Sr(A\)S?(AlA(A ®B),A®b\¡=i /

fn^(A¡)) A.AS/iflSHA',)) B,b). Dk

n\ \i=l / / \ \i=X / /

Suppose a = {ax, a2, ... , as} is a partition of N and imagine the associ-

ated node diagram. If the diagram is cut into two pieces along a vertical line

not containing any nodes then we obtain two new node diagrams each associ-

ated with a partition which in turn is associated with a character. We denote

the new partitions by a¡, the partition associated with the node diagram on

the left, and ar, the partition associated with the node diagram on the right.

We may now induce the tensor product of Xa and Xa to SN, thus obtaining

(Xa ®Xa)\ and investigate the possibility that there exists inequality between

Xa and (Xa ® Xa )T. Theorem 6 guarantees that

xn > (?T®X ).Theorem 6. Suppose a = {ax, a2, ...as} is a partition of N. Let p and t

be positive integers such that t < s, al > p for 1 < i < t, and p > a. for

t < j <s. Let

a, = {p ,a[+x,al+2, ... ,as} and ar = {ax-p, a2-p, ... , at-p}.

Then, (XaY, Y) > ((X ® Xa )TT, Y) for each decomposable Y £ TN .

Proof. We create a Young diagram Da by filling the first column of the a frame

with 1,2,... , s, the second column with 5 + 1, s + 2, ... etc. We consider

this tableaux to be the adjunction of two tableaux, one associated with a¡ and

containing the integers 1,2,... , N - « , where « = E'=i a¡ ~ tp , the other

associated with ar and containing 7V-«+l, N - n + 2, ... , N.

We let A, , A2, ... , As be the row sets of Da and, for 1 < i < s, A' =

A;n{l, 2, ... , N-n} and A' =Aixl{N-n+l, N-n + 2, ... , N}. Denoting

r by r and c by c we observe thata J a J

r = na;!^(A,),i=i

that

r (p!)'fl^(A¡) fj a,!^(A,.);=1 i=f+l


890

and

T. H. PATE

r = \$al-p)\A^(Ari)

i=i

where the tableaux associated with a¡ is obtained from Da by deleting the last

ax - p, columns, and the tableaux associated with ar is obtained from Da by

deleting the first p columns and then subtracting N - « from each entry in the

resulting tableaux.

Letting c denote ca and c" denote ca we observe that c = c ® c" and

that

K = (N\yXY<arca~l = (Nl\Ca$-xJ2°crco-Xa a

= (A/!|Cjr'E(T(c'®c'V(c/®c")c7"1.

Therefore, for decomposable Y £ TN we have

(XJ, Y) = K E (<r(c ® c")r(c ® c')o~XY, Y)o€SN

= KY,(r(c'®c')(Yx(o)®Y2(o)),(c'®c")(Yx(o)®Y2(o)))a

s Is

= K\\a^.Y[^î){cYx(o)®c'Y2(o)),cYx(o)®c'Y2(o)i=\ a \i=\

= KflailE(tlÂÂc^Ba),Aa®B\1 = 1 CT \i=l /

where K = (N\\Ca\)~x , Yx(a) and T2(cr) are as in Theorem 3,

Aa = ( FI ^(A,") ) c'Yx(o), and 5, = c"T2(ct).<i=t+x

Now, applying Theorem 5, we have

<lr,r)>K]>,!ni71=1 j = X

I ( I

Ot ,

n^(A¡) Aa,Aaw=l

X m^K) B^Bau'=l

í

= *(/>!)'£[(«,-/»)! II ̂ Eî^'*»'^'*»)1=1 ;'=/+! ff

x (V3c"72(ct), c"T2(ct))

where ^ = (%=x S^(A\))A , y,2 = (\[si=t+x ̂(A;))A , and ^ = (n|=1 ^(A^))v .

Since y/x y/2 = \p2\px , r = (p!)' rfj=<+i Q,!î ̂ 2 » and r„ = llLiK - ¿>)!v3 the



above is equal to

KYM'w'H*}' Yx(o))(c"rac"Y2(o), Y2(o)).CT

Letting % denote the set of all members of SN that increase on {1,2,... ,

N - «} and {N-n + l,N-n + 2,... , N}, and H denote

G({1, 2, ... , N- «}) • G({N-n + l,N-n + 2,...N}),

and applying Lemmas 2, 3, and 4 as well as formulas (2.6), (2.7), and (2.10),

we see that the last of the above expressions is

= ÊE(C\C'(tA)_171^)'(tA)~1)^V''(tV)"1F2(9')'(tV)"1F2(9'))

= *EE E Krîi-'^^.y.iîec^c^-^^.y^))9eVp€S„oesN_n

= K\Ca)\Ca\^'£(praca¡p-lY1(9), Yx(tp))(eraca6-xY2(tp), Y2(tp))

9 ft,8

^(XQYx(<p),Yx(tp))(XaY2(tp),Y2(tp))

= ((Xa¡®~Xa)Y,Y). D

For each partition a = {ax, a2, ... , as} we shall associate a derived parti-

tion a which, if t denotes the smallest positive integer such that at > a(+x, is

equal to {a, - 1, a2 - 1, ... , a( - 1, a(+1, a(+2 , ... , a$, l'} . The following

result, perhaps our most appealing in an aesthetic sense, follows immediately

from Theorems 3 and 6.

Theorem 7. If a is a partition and a is its derived partition then Xa > Xa,.

Proof. Let a = {ax,a2, ... , as} = {(p + I)', at+x, at+2, ... , as} where p >

al+x . Let a¡ = {p', a!+x , at+2, ... , as} and ar = {l'} . Letting <£ denote

(Xa ®Xa )T we have Xa > % by Theorem 6, and f > Xa, by Theorem 3.

Hence, Xa > Xa, as required. □

Successive application of Theorem 7 to a partition a yields the sequence

X > X , > X „ > X ,3) > • • • > X « = ea — a — a — a — — a '

where k = ax — 1 and e is the signum function. The corresponding sequence

of matrix inequalities is, for C £ %?N ,

dx (C) > dx ( (C) > dx „ (C) > ■ • ■ > det(C),

a dramatic improvement over Schur's result (1.3).

In a recent paper, see [10], Heyfron has shown that if a = {q + 1, iN~g~lj

and ß = {q, lN~q} then dx (C) > dx (C) for each C £ ßTN. Since ß = a

this result is merely a special case of Theorem 7.


892 T. H. PATE

But Theorems 4 and 6 applied separately give us a generalization of Heyfron's

result as well as an improvement on the per-det inequalities for the single-

hook immanents obtained by Merris and Watkins in [11]. Following [11] we

let Xk, for k = 1,2,... , «, denote the irreducible character associated with

{k, 1 } and we abbreviate dk with dk . Then Merris and Watkins have

shown that

dk(C)< E per(C[tp\tp])det(C(tp\tp))</>eQk.„

for each C £%An where k £ {1,2, ... , «} . Since the degree of Xk is (¡Jl}),

the above is equivalent to

dk(C)<(n/k)(nk) E ^r(C[tp\(p])det(C(tp\tp))

for each C £ ^n and k e {1,2,... ,«}. But direct application of Theorem

4 gives

¿k(c)<(nk) E per(C[tp\<p])det(C(tp\tp)),

hence

dk(C)<(k/n) E per(C[^])det(C(c*|p))

for each c6<^ and k £ {1,2, ... , «} .

Applying Theorem 6 to Xk+X , k £ {0, I, ... , n - 1}, with p = 1, t = 1,

and s = n - k we have a, = {1 } and ar = {k} so

dk+x(C)>(nkY E det(C[çi>|ç*])per(C(ç*|ç/>))

hence

^+1(C)>(£) E ^r(C[tp\tp])det(C(tp\tp)).

Combining this with the above we have

¿k(c)<(nk) E per(C[tp\tp])det(C(tp\<p))<dk+x(C)

for each Ce^ and k£{l,2,... ,«-l}.

To obtain the James-Li ebeck result, namely that if ß = {p, q, Y}, p > q

and p + q + r = N, then

per(C) > dx (C)ß

for each C G ß?N , we set a = {p + r, <?} and note that a{,) = {p + r-i, q, l'},

1 < /' < r. Hence, a(r) = ß, and

^(C) > dXi (C) > dXn (C) > • • • > rfx w (C)



for each C £ß?N. But, the author has shown in [9], that dx (C) < per(C) for

any 2-term partition a. We thus have the following strengthened version of the

James-Liebeck result:

oer(C)>dx(C)>dXai(C) > ■ ■■ >dx^(C) = dXß(C)

for C £ ßtN . But Theorem 7 gives even more since dx (C) >dx (C) for

1 < i < P + r - 2.

The partition a is obtained from a in the simplest manner by referring2 2

to the corresponding node diagram. For example, if a = {5 ,4,2} then the

node diagram is

To obtain the node diagram for a simply remove the last column of dots

from the above diagram and append it to the first column, thus obtaining

whose corresponding partition is {4 ,2 ,1 }. Continuing in this manner we

obtain a ={3,2 I5}, ..(3) {2\ r},and aw = {l18}

References

1. J. W. Neuberger, Norm of symmetric product compared with norm of tensor product, Linear

and Multilinear Algebra 2 (1974), 115-122.

2. T. H. Pate, A continuous analogue of the Lieb-Neuberger inequality, Houston J. Math. 12

(1986), 225-234.

3. I. Shur, Über endliche Gruppen and Hermitische Formen, Math. Z. 1 (1918), 184-207.

4. M. Marcus, On two classical results of I. Schur, Bull. Amer. Math. Soc. 70 (1964), 685-688.

5. _, The Hadamard theorem for permanents, Proc. Amer. Math. Soc. 15 (1964), 967-975.

6. E. H. Lieb, Proofs of some conjectures on permanents, J. Math. Mech. 16 (1966), 127-134.

7. M. Marcus, Finite dimensional multilinear algebra, Part II, Dekker, New York, 1975.

8. G. D. James and M. Liebeck, Permanents and immanents of Hermitian matrices, Proc.

London Math. Soc. (3) 55 (1987), 243-265.

9. T. H. Pate, Permanental dominance and the Soûles conjecture for certain right ideals in the

group algebra, Linear and Multilinear Algebra (to appear).

10. Peter Heyfron, Immanent dominance orderings for hook partitions, Linear and Multilinear

Algebra 24 (1988), 65-78.


894 T. H. PATE

U.R. Merris and W. Watkins, Inequalities and identities for generalized matrix functions,

Linear Algebra Appl. 64 (1985), 223-242.

12. M. A. Naimark and A. I. Stern, Theory of group representations, Grundlehren Math. Wiss.

246(1982).

13. J. P. Serre, Linear representations of finite groups, Graduate Texts in Math., Springer-Verlag,

New York, Heidelberg, and Berlin, 1987.

14. T. H. Pate, Generalizing the Fischer inequality, Linear Algebra Appl. 92 (1987), 1-15.

Department of Mathematics, 100 Mathematics Building, 231 West 18th Avenue,

Columbus State University, Columbus, Ohio 43210

Current address: Department of Mathematics, Auburn University, Auburn, Alabama, 36849


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PARTITIONS, IRREDUCIBLE CHARACTERS, AND INEQUALITIES