It is remarkable that some of these polynomials are expressed as a
specialization of the Schur functions.
Yablonskii-Vorob'ev polynomials are expressible by 2-core Schur
functions,
and Okamoto polynomials by 3-core Schur functions [7,8,16].
It is now recognized that these structures reflect the affine Weyl group
symmetry, as groups of the Bäcklund transformations [29].
The determinant formulas of Jacobi-Trudi type for Umemura polynomials of
P and P
resemble each other.
In both cases, they are expressed by 2-core Schur functions, and entries
of the determinant are given by the Laguerre
polynomials [5,17].
Furthermore, in a recent work,
it has been revealed that the entire families of the characteristic
polynomials for rational solutions of P,
which include Umemura polynomials for P
as a special case,
admit more general structures [12].
Namely, they are expressed in terms of the universal characters that are
a generalization of the Schur functions.
The latter are the characters of the irreducible polynomial
representations of
,
while the former were introduced to describe the irreducible rational
representations [11].
What kind of determinant structures do Umemura polynomials for
P admit?
Recently, Kirillov and Taneda have introduced a generalization of
Umemura polynomials for P
in the context of combinatorics and
have shown that their polynomials degenerate to the special polynomials
for P
in some limit [9,10].
This result suggests that the special polynomials associated with a
class of algebraic solutions to P
are also expressible by the
universal characters.
In this paper, we consider P
The aim of this paper is to investigate a class of algebraic solutions
to P (or S
) that originate from the fixed points
of the Bäcklund transformations corresponding to Dynkin automorphisms
and, then to present its explicit determinant formula.
Let us remark on the terminology of ``algebraic solutions''.
P admits several classes of algebraic solutions [1,2,3,13,14],
and the classification has not yet been established.
In this paper, we concentrate our attention to the above restricted
class of algebraic solutions.
This paper is organized as follows.
In Section 2,
we first present a determinant formula for a family of algebraic
solutions to P (or S
).
This expression is also a specialization of the universal characters,
and the entries of the determinant are given by the Jacobi polynomials.
The symmetry of P
is described by the affine Weyl group of
type
.
In Section 3, as a preparation for constructing special
solutions,
we present a symmetric description of Bäcklund transformations for
P
[6,20].
We also derive several sets of bilinear equations for the
-functions.
In Section 4,
starting from a seed solution on fixed points of a Dynkin automorphism,
we construct a family of algebraic solutions to P
(or S
) by application of Bäcklund transformations.
A family of special polynomials is extracted as the non-trivial factor
of the
-function,
and our algebraic solutions are expressed by a ratio of these
polynomials.
A proof of our result is given in Section 5.
As is well known,
P degenerates to P
, P
by
successive limiting procedures [25,4].
In Section 6,
we show that the family of algebraic solutions to P
given in
Section 2 degenerate to rational solutions to P
and
P
with the same determinant structures.
Section 7 is devoted to discussing the relationship to the
original Umemura polynomials for P
.