Last update: 27 June 2013
For the time being we shall work in an arbitrary but we shall use the notation of symmetric functions [Mac1979] rather than that of Thus for we shall write in place of for the exterior power, and in place of for the symmetric power of We have and if
Recall that if are partitions and the skew Schur function is defined by the formula
where It is zero unless
We generalize this definition as follows: let and let be partitions of length then the multi-Schur function is defined by
We also define
for any sequence of integers of length
Remark. In the definition (3.1) the argument is constant in each row of the determinant. We might therefore also define
with arguments constant in each column. However, we get nothing essentially new: if we define partitions by
where (so that and are the respective complements of and in the rectangle then we have
as one sees by replacing by in the determinant (3.1).
(3.2) We have
If then for some and hence
whenever and It follows that the matrix has an block of zeros in the south-west corner, and hence its determinant vanishes.
(3.3) If and then
We have for Hence for each the row of the matrix has zeros in the first places, and in the place.
An element is said to have finite rank if for all sufficiently large We then define the rank of to be the largest such that If both have finite rank, the formula
shows that has finite rank, and that
(3.4) Let with (so that Then for all
since if Hence the matrix
is the product of the matrix
and the matrix
which is unitriangular. Now take determinants.
So far the have been arbitrary elements of the But it seems that is mainly of interest when is an increasing sequence in in the sense that for
(3.5) Let be elements of each of rank and let
for each Then for all we have
In particular, if is a partition of length
From (3.4) we have
which is because
Hence the determinant at is triangular, with diagonal elements
The formula (3.5) now follows.
In particular, when all the are zero we have
for all Also, when all the are zero (and is a partition we have
If we replace the by and by this becomes
(3.6) Let be a partition of length and elements of a Suppose that are such that and
that is to say we can replace each by without changing the value of the multi-Schur function.
Let so that For all we have
It follows that if we replace the row of the determinant by
we shall obtain
with arguments equal to The proof is now completed by induction on
Let be a sequence in the such that
(3.7) Let be any interval in Then the inverse of the matrix
Let denote the matrix The element of is then
If this is zero; if it is equal to and if it is equal to
which is zero because
(3.8) (Duality Theorem, version) Let be partitions of length such that Then
Then the integers and fill up the interval and so do the and the
The minor of the matrix is
The complementary cofactor of has row indices and column indices Hence it is
Since each minor of is equal to the complementary cofactor of (because the result follows.
Remark. Observe that
Hence (3.8) gives a duality theorem for the multi-Schur function provided that for
At first sight the formula (3.8) is disconcerting, because the arguments on the left are not in general the negatives of the arguments on the right. However, we can use (3.6) to rewrite (3.8), as follows. As in Chapter I, let us write the partition in the form
where and each Then in
the first arguments are
which by (3.6) may all be replaced by where The next arguments are
which by (3.6) may all be replaced by where In general, for each the group of arguments may all be replaced by where Now if
is the conjugate partition, we have and is the content of the square in the diagram of The squares are the "salients" of the border of read in sequence from north-east to south-west. Hence the duality theorem (3.8) now takes the form
(3.8') (Duality Theorem, version). With the above notation, we have
Finally, if we set we have
Let now be independent indeterminates over We may regard as a by requiring that each has rank Let for each Then we have
Consider the generating functions: is the coefficient of in
in which the coefficient of is
The other two relations are proved similarly.
(3.10) Let and let If is such that for all then
where has coordinate equal to and all other coordinates zero.
By definition, we have and acts only on the row of the determinant, the entries in the other rows being symmetrical in and (because of the condition if Hence the first of the relations (3.10) follows from the first of the relations (3.9), and the other two are proved similarly.
Remark. We can use the relations (3.10) to give another proof of duality (3.8) in the form
(3.8''') Let be a partition such that and Then
Let be a corner square of the diagram of so that and Let be the partition obtained from by removing the square By operating on either side of with we obtain the same relation with replacing Hence it is enough to show that is true when but in that case both sides are equal to by (3.5'), (3.5'') and (3.6).
(3.11) Let be the longest element of Then for any we have
where and the are independent of
is a reduced word for so that
and likewise for By (3.10), applied to will produce
We have next to operate on this with which will produce
By repeating this process we shall obtain the formula for That for is proved similarly.
Remark. Let and in (3.11). Then by (3.5') we have
Thus we have independent proofs of (2.11) and (2.16') and hence (by linearity) of (2.10) and (2.16).
Let be independent variables and let
for all with the understanding that if and if
(3.12) (Sergeev) For all partitions we have
Let (resp. be the longest element of (resp. and let (resp. denote acting on the (resp. acting on the From (3.5) we have, if
and in view of (2.16) Sergeev's formula may be restated in the form
From (3.11) and (1) above we have
If (2) can be rewritten in the form
the duality theorem (3.8'') applies, and gives
where We can now apply (3.11) again and obtain from (3) and (4)
This is a typed excerpt of the book Notes on Schubert Polynomials by I. G. Macdonald.