Last update: 2 July 2013
Let be two sequences of independent indeterminates, and recall (5.8) that
For each we define the double Schubert polynomial to be
where acts on the variables.
Since we have
the (single) Schubert polynomial indexed by
From the Cauchy formula (5.10) we have
and by (4.2)
if i.e. if and
summed over all such that and
From (6.3) it follows that is a homogeneous polynomial of degree in We have
(i) is immediate from the definition (6.1).
(ii) and (iii) follow from (6.3).
(iv) follows from (5.20), since if
(6.5) (Stability) If and is the embedding of in then
This again follows from (6.3) and the stability of the single Schubert polynomials (4.5).
From (6.5) it follows that the double Schubert polynomials are well defined for all permutations
For any commutative ring let denote the of all functions on with values in We define a multiplication in as follows: for
summed over all such that and For this multiplication, is an associative (but not commutative) ring, with identity element the characteristic function of the identity permutation It carries an involution defined by
|(i)||If and is not a zero divisor in then|
|(iii)||is a unit (i.e. invertible) in if and only if is a unit in|
(i) We have and hence We shall show by induction on that for all So let and assume that for all such that Let be a permutation of length We have
where the sum on the right is over such that and so that and therefore Hence (1) reduces to and therefore as required.
(ii) We have so that is a unit in Also whence by (i) above.
(iii) Suppose is a unit in with inverse Since is we have whence is an unit in
Conversely, if is an unit in we construct an inverse of as follows. We define and proceed to define by induction on Assume that has been defined for all such that and set
summed over such that and This definition gives as required.
Now let (resp. be the function on whose value at a permutation is (resp. (The coefficient ring is now the ring of polynomials in the and Since it follows from (6.6)(iii) that and are units in
(i)-(iii) follow directly from (6.2) and (6.4).
From (6.3) and (6.4) we have
summed over such that and In other words,
In particular, when we obtain by (ii) above, and hence This establishes (iv); part (v) now follows from (iv) and (iii), and (vi) from (iv) and (1) above.
From (6.7) (vi) we have
summed over such that and so that and by (4.2). Hence
(where the operators act on the variables). The sum here may be taken over all permutations since unless By linearity and (4.13) it follows that
(6.8) (Interpolation Formula) For all we have
summed over permutations
(The reason for the restriction to in the summation is that if we shall have for some and hence where but for all since and therefore
Remarks. 1. By setting each in (6.8) we regain (4.14).
2. When the sum is over which consists of the permutations is dominant, of shape so that (see (6.15) below) Hence the case of (6.8) is Newton's interpolation formula
where or explicitly
For any integer let denote the polynomial obtained from by setting Since
where means it follows from the definitions (6.1) and (4.1) that
for all permutations Hence, by (6.7)(vi),
and in particular, for all integers
from which it follows that
Since is a sum of monomials with positive integral coefficients (4.17), is the number of monomials in (each monomial counted the number of times it occurs). By homogeneity, we have
From (6.7)(v) and (6.9) we obtain
so that we have another proof of the fact (4.30) that
Now consider the function whose value at is
For each positive integer we have
by (6.9). The value of (1) at a permutation of length is by (6.10) equal to
which is equal to (consider the coefficient of in On the other hand, is by definition equal to
summed over all sequences of permutations such that and for It follows that each has length hence say, and that is a reduced word for Since
by (4.4). we have and hence the sum (2) is equal to summed over all
We have therefore proved that
(6.11) The number of monomials in is
summed over all where
Remarks. 1. The reduced words for are where Hence from (6.11) and homogeneity we have
summed over as before. Letting we deduce that
2. If is dominant of length then is a monomial by (4.7). and hence in this case
3. Suppose that is vexillary of length Then by (4.9) we have
where is the shape of and the flag of Hence
for each If we now set each and then let we shall obtain in the limit the Schur function for the series ([Mac1979], Ch. I. §3. Ex. 5). which is equal to where is the product of the hook-lengths of Hence it follows from (6.12) that if is vexillary of length then
where is the shape of In other words. the number of reduced words for a vexillary permutation of length and shape is equal to the degree of the irreducible representation of indexed by
4. It seems likely that there is a of (6.11). Some experimental evidence suggests the following conjecture:
summed as in (6.11) over all reduced words for where
When is vexillary the double Schubert polynomial can be expressed as a multi-Schur function, just as in the case of (single) Schubert polynomials (Chap. IV). We consider first the case of a dominant permutation:
(6.14) If is dominant of shape then
where and for all
As in (4.6) we proceed by descending induction on The result is true for since is dominant of shape and
Suppose is dominant of shape Then (and Let be the largest integer such that for and let Then is dominant, and and therefore
by the inductive hypothesis; since it follows that
which is equal to by (3.5).
(6.15) If is Grassmannian of shape then
This follows from (6.14) just as (4.8) follows from (4.7).
Finally, let be vexillary with shape
as in Chapter IV. Then is also vexillary, with shape
the conjugate of and flag
where by (1.41)
With this notation recalled, we have
The proof is essentially the same as that of (4.9) (which is the case By (4.10) the dominant permutation constructed from in the proof of (4.9) has shape
and therefore by (6.15) we have
where and the sequence is obtained by subtracting the sequence term by term from the sequence Hence the same argument as in (4.9) establishes (6.17).
Remark. From (6.16) and (6.4)(iii) we obtain
where so that (if for each
by (1.41). Hence (6.4)(iii) reduces to the duality theorem (3.8'') (with when is vexillary.
Let (resp. be the shift operator (4.21) acting on the (resp. variables. Then we have
for all and all permutations
By (6.3) and (4.21) we have
summed over such that and By (6.3) again, the right-hand side is equal to
In particular, suppose that is vexillary. With the notation of (6.16), the flag of (resp. is obtained from that of (resp. by replacing each by (resp. each by Hence by (6.16) we have
for all where (resp. is the homomorphism of (4.25) acting on the (resp. variables.
(6.19) Let (resp. denote acting on the (resp. variables. Then if is vexillary of shape we have
By (4.24) we have and Hence (6.19) follows from (6.17) and (6.18).
In particular, suppose that is dominant of shape so that by (6.14)
In this case (6.19) gives
for all which is Sergeev's formula (3.12').
This is a typed excerpt of the book Notes on Schubert Polynomials by I. G. Macdonald.