Page 1
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
Page 2
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.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 3
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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 4
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~'*>
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 5
GENERALIZED MATRIX FUNCTIONS 879
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 k^O then
(2.5) ^rj^ff"1 = ^o%yio~x = k~x^a%yi%o~x.
a a a
Proof. If f £ CSn and x £ Sn then, by setting u = xo~x, one sees that
^ofxo~x = Y^u~Xxfu = ^oxfo~X,a u a
an equality that immediately implies that if /, g £ CSn then
^ofgo~x =^2ogfo~x.ct a
Consequently, we have
Y,oy/%o~x = k~x^oyi%2o~x = k~l ^ogy/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.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 6
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„
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 7
GENERALIZED MATRIX FUNCTIONS 881
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 = ^o(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/ < §? .
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 8
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)
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 9
GENERALIZED MATRIX FUNCTIONS 883
for each q £ Tn+pm . Hence,
ii2\W(A)(A®B)\\2= E E \^(^)(A®Bt)(eq)\2
'err,mî€r„+p m
= E II^A)^®^)!!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,
+ *) H^A,)^!!2. E IK(A2)v5r|« +P
'6rr
"+P) |K(A^||V(A2)V¿?||2. D
Given y = y* ® y2 ® ■ • ■ ® y^ where y, , y2 , ... .y^e 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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 10
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)\(\RJ\Ca\)-X^orc"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)\(n+nP\ K^\\jt/(A')((c"r)AYx(o))\\2\\sf(A0)Y2(o)\\2^ ' CT
by Theorem 2, where A' = A, - {N - « + 1, N - « + 2, ... , N} and A0 =
{1,2,... ,«}. Letting % be as in the statement of the theorem and observing
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 11
GENERALIZED MATRIX FUNCTIONS 885
that s/(A')c" = (p\)~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!)-l^EEiK'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!)-^EE<ctt'ra'M *»' c*'ra'u Yx(tp))\\^(AQ)e-xY2(tp)\\¿
</> u,e
= n\(p\)-X\Ca,\KY,Y.^ura'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 = l^a'lE^a'^'^"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) E^a'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,...
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 12
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
= E^i ,'••• '*« >>*«, ,'■•■ '*« )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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 13
GENERALIZED MATRIX FUNCTIONS 887
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\(=1
> n^(4)\i=X
f k \v
n^K) b\i=X J
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 14
T. H. PATE
for each A £ TN and B e Tp. Equivalently,
U[^)(jl^{ài)(A®B),A®B\
Proof. Let A¿ = {Sn,S¡2,... ,Sjm} where we assume SiX < Si2 < ••• <
àim., 1 < i < k. For 1 < j < c¡, c; = minf«;, p(.}, we let yi} denote
ri/=i(^M ' a¡ s+n), a product of disjoint transpositions. Now, by Lemma 6,
n(:0iVw=n it ( nj ){pj) ^(a;)^(a>,^(a;)^(a;)
- E E ■ • • E ft ( /' ) iPi ) ̂ m^y^A])^).j.=oj2=o a=0/=i Wi/ V7'/
Given A £ TN and B £ Tp we have
k
n^(A;.)^(A>u. ^(a;)^(a;.) j (¿ ® b) , a ® 5
■((S^)(ffiHA"(5Hv')"
=/[n^)^'®5')^'®Ä')
where Ä = (Y[U^(A\))aA and B' = (YlU^i^)) B ■ But [&, fy » aproduct of disjoint transpositions (rs) such that 1 < r < N and N + I < s <
N + P . Therefore,
((n^,)(^®5')^'®5')>°
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 15
GENERALIZED MATRIX FUNCTIONS 889
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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 16
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[^^i){cYx(o)®c'Y2(o)),cYx(o)®c'Y2(o)i=\ a \i=\
= KflailE(tl^A^Ac^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 ̂ E^i^'*»'^'*»)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,!^i ̂ 2 » and r„ = llLiK - ¿>)!v3 the
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 17
GENERALIZED MATRIX FUNCTIONS 891
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
= ^EE(C\C'(tA)_171^)'(tA)~1)^V''(tV)"1F2(9')'(tV)"1F2(9'))
= *EE E Kr^ii-'^^.y.i^iec^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.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 18
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)
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 19
GENERALIZED MATRIX FUNCTIONS 893
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.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
Page 20
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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use