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Derivation of the eigenvalues of the configuration process induced by a labeled direct product branching process
THEORETICAL
POPULATION
BIOLOGY
7, 221-228 (1975)
Derivation of the Eigenvalues Induced by a Labeled Direct
of the Configuration Product Branching
Process Process*
SAMUEL KARLIN Department
of Theoretical Mathematics,
The Weizmann Institute of Science, Rehovot, Israel, and
Mathematics
Department,
Stanford
University,
Stanford,
California
AND HAIM AVNI Department
of Theoretical
Mathematics,
The Weizmann Institute of Science, Rehovot, Israel
Ewens and Kirby in the previous paper determined the full set of eigenvalues in the (Wright) neutral alleles finite population model admitting an infinite number of possible allelic types Q where each gene mutation creates a novel type (see their paper for background discussion of the problem and other references). They use a method akin to that of identity by descent and some combinatorial devices. In this note the eigenvalues of the configuration process (defined precisely later) induced by a direct product branching process (which includes the Wright model) (Karlin and McGregor (1964); see also Karlin (1966, p. 416)) are derived from knowledge of the eigenstructure set forth in Karlin and McGregor (1966) on the labeled process, combined with an elementary symmetrization procedure. For completeness, we review the formulation and key facts needed, (see (3, pp. 396-416)). Consider a haploid population of N individuals composed from possible allelic types A, , A, ,..., A, . A state in the population process is described by a p-tuple, i = (ix , iZ ,..., i,) where i, is the number of occurrences of allele A, . Of course xF=, iV = N. The collection of all states is denoted by d. Suppose each individual of type A, in one generation produces progeny of all types according to the probability generating function fk(sl , s2,-., s,) =
1 aS~~vz,...,vps3y22 *.. ~2, Y&O
h = 1) 2 )...) p,
* This paper was inspired by Ewens and Kirby (1975). We appreciate them making available their preprint. 221 Copyright AI1 rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
222
KARLIN AND AVNI
where4$a,...,Y is the probability that one parent of type A, will produce offspring consistkg of or of type A, , v2 of type A,, etc. Individuals are assumed to act independently. Let (X1(n), X2(n),..., X,(n)) denote the associated branching process, where X,(n) represents the number of A, types at the start of the nth * ’ generating function of the progeny after one generation. Th e probablhty generation is f>(s, )...) s,)&sl
,..., SD)-.fk(sl
)..., SD)h(s, )...) sp),
where X,(O) = iI, (k = 1, 2,...,p) and h(s, ,..., sD) denotes the generating function of the various types immigrating into the system. We construct the transition probability matrix to be X,(O) = i, ) p = I,...) p,
x,(l)=j,,v=l,...,
p
’ 2 X(O) = i "Zl
=
coeff s”;IsZj.I. s> infi1(s)f12(s) coeff tN in l-J:=, Q(t)
X”(1) = N
(1)
"=l
. ..f$(s) h(s) h(t)
where s = (sl ,..., sJ, t = (t ,..., t). In the special case ii(s) = exp /ui (f
qs,, - I)/
"=l
(2)
% 3 0,
c aiv = 1,
ui > 0,
c, 3 0,
V) i = l)..., p,
the transition probability matrix (1) reduces to
The parameters occurring in (3) are to be interpreted as follows. a,,, represents the chance that an A, type will mutate after birth into an A, type; cr, measures the relative fitness coefficient of an A,, type; and c, is the average rate at which A, types are immigrating into the system.
EIGENVALUES
OF CONFIGURATION
223
PROCESS
We specialize now to h(s) = 1 (no immigration) and where
(4) where f(s) is independent of Y (reflecting no selection differences, that is, each parental type produces progeny following the same p.g.f. f(s) after which mutations may occur). Let r = I/ olypijzU.=i denote the mutation matrix and assume the eigenvalues of r are 1 = y1 9y2 7.-e,y1, ,
IYil < 1, 2
with a full set of corresponding eigenvectors u(l), u(~),..., u(p) available. Then a complete set of eigenvalues for 11Pi,i ]I consists of A, = 1,
and
2 < qi
~Y,~Y,, ... yq, T
(54
where h = r
coeff sN+ inf”+(s)[f’(s)]r coeff sN in f”(s)
*
(5b)
In the special case of the Wright process (3) (without migration pressures), we have A, = 1,
A, = (I -
l/N)(l
-
2/N) *.. (1 - (r -
1)/N),
l
In order to characterize the associated eigenvectors to (5a), we define the sequence of linear functions
L*(x) = L&c, )...)x,)= f uyx,)
q = 2,..., p.
v=l
Because {u(*)} are linearly independent, so are the linear forms L, . It is proved in Karlin (1966) that an eigenvector a,l,,..,,l associated with &.YQJ~, **. yp, has its i E A coordinate a,,....,&)
= LJi)L,Ji)
wherePQ1,...J.)is an appropriate
...L,ji)
+ P,,....,&
(7)
polynomial of degree < Y. The vectors (7) span a complete set of right eigenvectors for j/ Piei 11,referred to hereafter as the labeled direct product branching process. We have now recorded the general formulation and some relevant results on the direct product branching model for thep-allele labeled process. (The analysis to follow carries over also for the overlapping generation symmetric haploid models studied by Cannings (1974).)
224
KARLIN
AND
AVNI
We next specialize to a case where the mutation process is symmetric in its allelic e#ects. Specifically, we prescribe the transition matrix for mutation events to be OLij
=
[U/(
p
-
I)]
+
(1
-
U)
(& = Kronecker delta symbol),
sij
i,j = 1,2 ,...) p,
(8)
such that u is the probability of a mutation event per individual/per generation and the change is equally likely to any one of the other types. The eigenvalues of the specific mutation matrix r = 11olii I/ of (8) are explicitly with simple multiplicity having multiplicity p - 1.
I’, (1
-
[Pi’(P
-
I)]%
(9)
A symmetrical listing of right eigenvectors of r corresponding to the eigenvalue 1 - [ p/( p - I)]u consists of u(Q)
=
where
(&I) 1
,...,
$,,
~2’ = fl,l,(p
_ *)
3
i g i
q = 1, 2, 3)..,) p.
> (10)
Of course, only p - 1 among {u*},” are linearly independent. In the circumstances of (8), we see on the basis of (9) that the eigenvalues of (1Pi.j (1have the form
(1 - UP - l>/Pl4% 9
Y = 0, l,...,
(11)
occurring with multiplicity ( ‘+F-‘) where &. is defined in (5b). With the aid of the facts connected with (7) and because of the special symmetric form of (8) we find that all eigenvectors admit the following characterization. With each specification of a p-tuple of integers, q = (ql , qz ,..., qD) satisfying 0 < Q” >
v = 1) 2, .... p,
and
iqv
=r,
(12)
we define L,(e) = L,(.$, ,..., f,) = zfzl uy)& , q = 1, 2 ,..., p and then each eigenvector of 11Pi,) /I for (1 - [(p - l)/p]u)‘h, has the representation b(c,~.az-,~p)(i) = fi [~y(i>l”” + P,(i),
(13)
v=l
where P,, is an appropriate polynomial in p real variables satisfying deg P, < r. Because of the special form of u(q) given in (10) we actually have at the i = (6 ,..., i,) component b,(i) = ip . ip ... i2 + p,,(i),
with
deg(pa) < Y.
(14)
EIGENVALUES
OF CONFIGURATION
225
PROCESS
For the specification of (8) and (9) all alleles behave symmetrically in all senses and we can construct an associated configuration process to the direct product branching process eliminating the dependence on the labeling assigned to alleles. We proceed to this construction. A configuration state is a collection of states i E d characterized by a prescription of integers (K, , K, ,..., K,), m < p, C k, = N where one of the alleles (without saying which) is represented k, times, a second k, times,..., and an mth allele k, times. Consider the set Y of aZZpermutations T mapping (1, 2,..., p) into itself corresponding to all relabelings of the alleles. We define r applied to i to give vi = (Cdl) , id2) ,..., id,)). Thus, the collection of states obtained from ni with n traversing Y consists of all states where some allele appears ir times, another allele appears is times, etc., without specifying the labelings of the alleles. Consider v;y
pi,dt)
=
8i.1
-
(15)
Because of the symmetry in the mutation process it is easy to check that for all
7r0
and
TlEYP,
(16)
and accordingly B,,r is the transition probability matrix defined on the configuration space (unlabeled process). Our objective is to determine the eigenvalues of the configuration process. Consider an eigenvector b,(i) (described in (14)) for the eigenvalue pr = (1 - [ p/( p - l)]u]‘& . We claim that for any element m E 9 the vector b,V)
iEA,
= b&G)),
(17)
is also an eigenvector for the same eigenvalue. Indeed, since b,(i) is an eigenvector, we have
A change of variable j to rj and since P,,i,,, = P,,, (due to the inherent symmetry of mutation and reproduction events) the claim is validated. The symmetrized sum of eigenvectors
is manifestly satisfies
also an eigenvector for CL,.= (I - [p/( p 6,(i) = b,(A),
for all
77E Y.
I)]Q)$.
and bq (19)
226
KARLIN
AND AVNI
Therefore, 6, can be interpreted as an eigenvector of tar for the configuration process. It is also clear that 6, is a symmetric polynomial in the components of i. The next familiar combinatorial quantity is of relevance (cf. Ewens and Kirby (1975)). DEFINITION 1. Set m(0) = 1. Let m(r) for r > 1 be the number of distinct partitions of r into positive integers constrained such that each contribution to the partition has value at least 2. To illustrate, we have m(l) = 0, m(2) = 1, m(3) = 1, m(4) = m(5) = 2, m(6) = 4. The configuration space %’ contains M(V) states which can be parameterized in a 1: 1 manner as the collection of all distinct decreasing sequences
i = (il , i2 , . . . , Q,
where
i, 2 iz > *a. > i, >, 0,
This number can be represented in the form M(Q) = F m(r)
(providedp
2 N)
(21)
7=1
as may be validated by decomposing V according to the number of I’s in the decreasing sequences, i.e., %?= u+r Vr, where %r = {(ir ,..., &) ( ik 3 2 for k < 1, CkGl ik = r, ik < 1 for k > I}, The next theorem is basic. The number of linearly process for the eigenvalue THEOREM.
independent
pr = (1- [ $4P - lll4f A,ism(r)
vectors in the configuration
(see Dejnition
1).
The proof will be best elucidated by splitting it into two elementary lemmas. First, it is convenient to introduce the polynomial leading part of 6,(e). Accordingly, for each set of integers, q = (q1 ,***, qN) satisfying qi 3 2 or qi = 0, qi 3 qi+l , i = 1, Z..., N, and N
C qi = r,
(22)
i=l
we form P,(i)
=
C fi ii;,, , 7627
"4
(23)
EIGENVALUES
OF CONFIGURATION
PROCESS
227
and the factorial polynomials (24) interpreted by the convention x(j) = X(X - 1) *em(X - j + 1). LEMMA 1. The r’, of (24) deji ned on the state space A with q fuljilling the constraints (22) can be expressedas a linear combination of the P,, (of (23)) over A with q subject to the same constraints.
Proof. Recall the fact that every symmetric polynomial in the variables (4 , 4 ,..., i,) can be written as a linear combination of polynomials in the variables ok = ilk + izk + .*- + ipk,
h = 0, 1, 2 ,...,
and because or = i, + is + .*. + i, = N is a constant on A the result of the lemma is clear. LEMMA 2. Arranging i = (iI ,..., i,) E A and q = (ql , q2 ,..., qD) (q satisfying the constraints of (22)) in lexicographic order, the matrix array 11pq(i)\\,,, is triangular with positive terms on the diagonal.
Proof.
Observe first that for q = (qr , us ,..., qr , 0,O .*a0)
E&l1 >***,42> l,..., 1, 0 ..* 0) = @)qF) *** &I) + other nonnegative terms 2 41! a! .** qr! > 0. Moreover, if q > i in the sense of the lexicographic ordering, then each of the products 47,)+?.J i(Og’ zd1)z7-r(2) *.-
n(Z)
contains at least one factor of the form i(g) where Q > i which is zero. This proves the lemma. Proof of the theorem. By Lemmas 1 and 2 we are guaranteed, by the independence of the leading parts P, of the 6, , that the eigenvalue 11.rhas at least m(r) linearly independent eigenvectors. Since &, m(r) = M(g) = number of states in the configuration space (21), we conclude that there are exactly m(r) linearly independent eigenvectors associated with the eigenvalue pr = (1 - [ p/( p - l)]u)‘h, and the proof of the theorem is complete. The result of Ewens and Kirby is obtained by sending p -+ co. Similar results can be enunciated for symmetric migration patterns in place of the mutation process (8). 65317124
228
KARLIN
AND AVNI
Remark. The probability expressions in Ewens and Kirby actually also symmetric polynomials over A.
(1975) are
REFERENCES EWENS, W. AND KIRBY, K. 1975. The eigenvalues of the neutral alleles process, Pop. Biol., to appear. KARLIN, S. AND MCGREGOR, J. L. 1964. Direct product branching processes and Markov chains, Proc. Nat. Acad. Sci. 4, 598-602. KARLIN, S. 1966. “A First Course in Stochastic Processes,” Academic Press, New CANNINGS, C. 1974. The latent roots of certain Markov chains arising in genetics: approach. I. Haploid models, Adv. Appl. Prob.
Theor. related York. A new
POPULATION
BIOLOGY
7, 221-228 (1975)
Derivation of the Eigenvalues Induced by a Labeled Direct
of the Configuration Product Branching
Process Process*
SAMUEL KARLIN Department
of Theoretical Mathematics,
The Weizmann Institute of Science, Rehovot, Israel, and
Mathematics
Department,
Stanford
University,
Stanford,
California
AND HAIM AVNI Department
of Theoretical
Mathematics,
The Weizmann Institute of Science, Rehovot, Israel
Ewens and Kirby in the previous paper determined the full set of eigenvalues in the (Wright) neutral alleles finite population model admitting an infinite number of possible allelic types Q where each gene mutation creates a novel type (see their paper for background discussion of the problem and other references). They use a method akin to that of identity by descent and some combinatorial devices. In this note the eigenvalues of the configuration process (defined precisely later) induced by a direct product branching process (which includes the Wright model) (Karlin and McGregor (1964); see also Karlin (1966, p. 416)) are derived from knowledge of the eigenstructure set forth in Karlin and McGregor (1966) on the labeled process, combined with an elementary symmetrization procedure. For completeness, we review the formulation and key facts needed, (see (3, pp. 396-416)). Consider a haploid population of N individuals composed from possible allelic types A, , A, ,..., A, . A state in the population process is described by a p-tuple, i = (ix , iZ ,..., i,) where i, is the number of occurrences of allele A, . Of course xF=, iV = N. The collection of all states is denoted by d. Suppose each individual of type A, in one generation produces progeny of all types according to the probability generating function fk(sl , s2,-., s,) =
1 aS~~vz,...,vps3y22 *.. ~2, Y&O
h = 1) 2 )...) p,
* This paper was inspired by Ewens and Kirby (1975). We appreciate them making available their preprint. 221 Copyright AI1 rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
222
KARLIN AND AVNI
where4$a,...,Y is the probability that one parent of type A, will produce offspring consistkg of or of type A, , v2 of type A,, etc. Individuals are assumed to act independently. Let (X1(n), X2(n),..., X,(n)) denote the associated branching process, where X,(n) represents the number of A, types at the start of the nth * ’ generating function of the progeny after one generation. Th e probablhty generation is f>(s, )...) s,)&sl
,..., SD)-.fk(sl
)..., SD)h(s, )...) sp),
where X,(O) = iI, (k = 1, 2,...,p) and h(s, ,..., sD) denotes the generating function of the various types immigrating into the system. We construct the transition probability matrix to be X,(O) = i, ) p = I,...) p,
x,(l)=j,,v=l,...,
p
’ 2 X(O) = i "Zl
=
coeff s”;IsZj.I. s> infi1(s)f12(s) coeff tN in l-J:=, Q(t)
X”(1) = N
(1)
"=l
. ..f$(s) h(s) h(t)
where s = (sl ,..., sJ, t = (t ,..., t). In the special case ii(s) = exp /ui (f
qs,, - I)/
"=l
(2)
% 3 0,
c aiv = 1,
ui > 0,
c, 3 0,
V) i = l)..., p,
the transition probability matrix (1) reduces to
The parameters occurring in (3) are to be interpreted as follows. a,,, represents the chance that an A, type will mutate after birth into an A, type; cr, measures the relative fitness coefficient of an A,, type; and c, is the average rate at which A, types are immigrating into the system.
EIGENVALUES
OF CONFIGURATION
223
PROCESS
We specialize now to h(s) = 1 (no immigration) and where
(4) where f(s) is independent of Y (reflecting no selection differences, that is, each parental type produces progeny following the same p.g.f. f(s) after which mutations may occur). Let r = I/ olypijzU.=i denote the mutation matrix and assume the eigenvalues of r are 1 = y1 9y2 7.-e,y1, ,
IYil < 1, 2
with a full set of corresponding eigenvectors u(l), u(~),..., u(p) available. Then a complete set of eigenvalues for 11Pi,i ]I consists of A, = 1,
and
2 < qi
~Y,~Y,, ... yq, T
(54
where h = r
coeff sN+ inf”+(s)[f’(s)]r coeff sN in f”(s)
*
(5b)
In the special case of the Wright process (3) (without migration pressures), we have A, = 1,
A, = (I -
l/N)(l
-
2/N) *.. (1 - (r -
1)/N),
l
In order to characterize the associated eigenvectors to (5a), we define the sequence of linear functions
L*(x) = L&c, )...)x,)= f uyx,)
q = 2,..., p.
v=l
Because {u(*)} are linearly independent, so are the linear forms L, . It is proved in Karlin (1966) that an eigenvector a,l,,..,,l associated with &.YQJ~, **. yp, has its i E A coordinate a,,....,&)
= LJi)L,Ji)
wherePQ1,...J.)is an appropriate
...L,ji)
+ P,,....,&
(7)
polynomial of degree < Y. The vectors (7) span a complete set of right eigenvectors for j/ Piei 11,referred to hereafter as the labeled direct product branching process. We have now recorded the general formulation and some relevant results on the direct product branching model for thep-allele labeled process. (The analysis to follow carries over also for the overlapping generation symmetric haploid models studied by Cannings (1974).)
224
KARLIN
AND
AVNI
We next specialize to a case where the mutation process is symmetric in its allelic e#ects. Specifically, we prescribe the transition matrix for mutation events to be OLij
=
[U/(
p
-
I)]
+
(1
-
U)
(& = Kronecker delta symbol),
sij
i,j = 1,2 ,...) p,
(8)
such that u is the probability of a mutation event per individual/per generation and the change is equally likely to any one of the other types. The eigenvalues of the specific mutation matrix r = 11olii I/ of (8) are explicitly with simple multiplicity having multiplicity p - 1.
I’, (1
-
[Pi’(P
-
I)]%
(9)
A symmetrical listing of right eigenvectors of r corresponding to the eigenvalue 1 - [ p/( p - I)]u consists of u(Q)
=
where
(&I) 1
,...,
$,,
~2’ = fl,l,(p
_ *)
3
i g i
q = 1, 2, 3)..,) p.
> (10)
Of course, only p - 1 among {u*},” are linearly independent. In the circumstances of (8), we see on the basis of (9) that the eigenvalues of (1Pi.j (1have the form
(1 - UP - l>/Pl4% 9
Y = 0, l,...,
(11)
occurring with multiplicity ( ‘+F-‘) where &. is defined in (5b). With the aid of the facts connected with (7) and because of the special symmetric form of (8) we find that all eigenvectors admit the following characterization. With each specification of a p-tuple of integers, q = (ql , qz ,..., qD) satisfying 0 < Q” >
v = 1) 2, .... p,
and
iqv
=r,
(12)
we define L,(e) = L,(.$, ,..., f,) = zfzl uy)& , q = 1, 2 ,..., p and then each eigenvector of 11Pi,) /I for (1 - [(p - l)/p]u)‘h, has the representation b(c,~.az-,~p)(i) = fi [~y(i>l”” + P,(i),
(13)
v=l
where P,, is an appropriate polynomial in p real variables satisfying deg P, < r. Because of the special form of u(q) given in (10) we actually have at the i = (6 ,..., i,) component b,(i) = ip . ip ... i2 + p,,(i),
with
deg(pa) < Y.
(14)
EIGENVALUES
OF CONFIGURATION
225
PROCESS
For the specification of (8) and (9) all alleles behave symmetrically in all senses and we can construct an associated configuration process to the direct product branching process eliminating the dependence on the labeling assigned to alleles. We proceed to this construction. A configuration state is a collection of states i E d characterized by a prescription of integers (K, , K, ,..., K,), m < p, C k, = N where one of the alleles (without saying which) is represented k, times, a second k, times,..., and an mth allele k, times. Consider the set Y of aZZpermutations T mapping (1, 2,..., p) into itself corresponding to all relabelings of the alleles. We define r applied to i to give vi = (Cdl) , id2) ,..., id,)). Thus, the collection of states obtained from ni with n traversing Y consists of all states where some allele appears ir times, another allele appears is times, etc., without specifying the labelings of the alleles. Consider v;y
pi,dt)
=
8i.1
-
(15)
Because of the symmetry in the mutation process it is easy to check that for all
7r0
and
TlEYP,
(16)
and accordingly B,,r is the transition probability matrix defined on the configuration space (unlabeled process). Our objective is to determine the eigenvalues of the configuration process. Consider an eigenvector b,(i) (described in (14)) for the eigenvalue pr = (1 - [ p/( p - l)]u]‘& . We claim that for any element m E 9 the vector b,V)
iEA,
= b&G)),
(17)
is also an eigenvector for the same eigenvalue. Indeed, since b,(i) is an eigenvector, we have
A change of variable j to rj and since P,,i,,, = P,,, (due to the inherent symmetry of mutation and reproduction events) the claim is validated. The symmetrized sum of eigenvectors
is manifestly satisfies
also an eigenvector for CL,.= (I - [p/( p 6,(i) = b,(A),
for all
77E Y.
I)]Q)$.
and bq (19)
226
KARLIN
AND AVNI
Therefore, 6, can be interpreted as an eigenvector of tar for the configuration process. It is also clear that 6, is a symmetric polynomial in the components of i. The next familiar combinatorial quantity is of relevance (cf. Ewens and Kirby (1975)). DEFINITION 1. Set m(0) = 1. Let m(r) for r > 1 be the number of distinct partitions of r into positive integers constrained such that each contribution to the partition has value at least 2. To illustrate, we have m(l) = 0, m(2) = 1, m(3) = 1, m(4) = m(5) = 2, m(6) = 4. The configuration space %’ contains M(V) states which can be parameterized in a 1: 1 manner as the collection of all distinct decreasing sequences
i = (il , i2 , . . . , Q,
where
i, 2 iz > *a. > i, >, 0,
This number can be represented in the form M(Q) = F m(r)
(providedp
2 N)
(21)
7=1
as may be validated by decomposing V according to the number of I’s in the decreasing sequences, i.e., %?= u+r Vr, where %r = {(ir ,..., &) ( ik 3 2 for k < 1, CkGl ik = r, ik < 1 for k > I}, The next theorem is basic. The number of linearly process for the eigenvalue THEOREM.
independent
pr = (1- [ $4P - lll4f A,ism(r)
vectors in the configuration
(see Dejnition
1).
The proof will be best elucidated by splitting it into two elementary lemmas. First, it is convenient to introduce the polynomial leading part of 6,(e). Accordingly, for each set of integers, q = (q1 ,***, qN) satisfying qi 3 2 or qi = 0, qi 3 qi+l , i = 1, Z..., N, and N
C qi = r,
(22)
i=l
we form P,(i)
=
C fi ii;,, , 7627
"4
(23)
EIGENVALUES
OF CONFIGURATION
PROCESS
227
and the factorial polynomials (24) interpreted by the convention x(j) = X(X - 1) *em(X - j + 1). LEMMA 1. The r’, of (24) deji ned on the state space A with q fuljilling the constraints (22) can be expressedas a linear combination of the P,, (of (23)) over A with q subject to the same constraints.
Proof. Recall the fact that every symmetric polynomial in the variables (4 , 4 ,..., i,) can be written as a linear combination of polynomials in the variables ok = ilk + izk + .*- + ipk,
h = 0, 1, 2 ,...,
and because or = i, + is + .*. + i, = N is a constant on A the result of the lemma is clear. LEMMA 2. Arranging i = (iI ,..., i,) E A and q = (ql , q2 ,..., qD) (q satisfying the constraints of (22)) in lexicographic order, the matrix array 11pq(i)\\,,, is triangular with positive terms on the diagonal.
Proof.
Observe first that for q = (qr , us ,..., qr , 0,O .*a0)
E&l1 >***,42> l,..., 1, 0 ..* 0) = @)qF) *** &I) + other nonnegative terms 2 41! a! .** qr! > 0. Moreover, if q > i in the sense of the lexicographic ordering, then each of the products 47,)+?.J i(Og’ zd1)z7-r(2) *.-
n(Z)
contains at least one factor of the form i(g) where Q > i which is zero. This proves the lemma. Proof of the theorem. By Lemmas 1 and 2 we are guaranteed, by the independence of the leading parts P, of the 6, , that the eigenvalue 11.rhas at least m(r) linearly independent eigenvectors. Since &, m(r) = M(g) = number of states in the configuration space (21), we conclude that there are exactly m(r) linearly independent eigenvectors associated with the eigenvalue pr = (1 - [ p/( p - l)]u)‘h, and the proof of the theorem is complete. The result of Ewens and Kirby is obtained by sending p -+ co. Similar results can be enunciated for symmetric migration patterns in place of the mutation process (8). 65317124
228
KARLIN
AND AVNI
Remark. The probability expressions in Ewens and Kirby actually also symmetric polynomials over A.
(1975) are
REFERENCES EWENS, W. AND KIRBY, K. 1975. The eigenvalues of the neutral alleles process, Pop. Biol., to appear. KARLIN, S. AND MCGREGOR, J. L. 1964. Direct product branching processes and Markov chains, Proc. Nat. Acad. Sci. 4, 598-602. KARLIN, S. 1966. “A First Course in Stochastic Processes,” Academic Press, New CANNINGS, C. 1974. The latent roots of certain Markov chains arising in genetics: approach. I. Haploid models, Adv. Appl. Prob.
Theor. related York. A new