[seqfan] playing with Schur functions

[Wouter Meeussen]

I try to report some ‘pretty’ integer math games bases on the behaviour of Schur functions. It will result in a set of integer matrices with (apparently) zero determinant, cute symmetries, non-negative integer eigenvalues and integer eigenvectors. The eigenvalues can (apparently) be written in a very simple form.
[[ essentially same as I. G. Macdonald , Symmetric_Functions_and_Hall_Polynomials, eqn 5.9 ]]

Notation:
partitions : read “la” as “lambda”, “mu” as “mu”, “ze” as “zeta” ; partitions are ordered in reverse lexicographic order;
Schur functions : read as Schur polynomials in sufficiently many variables to avoid loss of info.
“appears to, apparently” : read as : checked up to n=11, proof left as an excercise to the reader.



What happens if you take a skew Schur function s_la/ze and multiply it by s_ze ?
--------------------------------------------------------------------------------------------------------

if partition la (stands for lambda) has weight n, and partition ze (for zeta) has weight k<n, then the skew Schur function expands in Schur functions indexed by partitions of (n-k). Multiplying by s_ze again makes the product expand into Schur functions of weight n again.
The coefficients are given by the respective (products of) LR-coefficients.

These same coefficients can be denoted in shorthand by the in-product < s_mu , s_ze s_la/ze > where la and mu are partitions of n and ze is a partition of k<n such that la majorizes ze (or the Young tabloid of la covers that of ze completely).

Example 1: 
la={3,1}, ze={1}
s_{3,1}/{1} = s_{3}+s_{2,1} 
multiply back by s_ze gives (s_{3}+s_{2,1}) s_{1} = s_{4} +2s_{3,1} +s_{2,2} +s_{2,1,1}
so the coefficient of s_mu=s_{3,1} is 2, or
<s_{3,2} , s_{3,1}/{1} s_{1} > = 2

Now look at the matrices of these in-products,
where we put the partitions la by row and mu by column, P(n) of them of course,
given by

T(k)_la_mu = Sum over <s_mu , s_la/ze s_ze >  where the sum runs over all partitions ze of weight k<n that are majorised by la, 
and admire their properties:

T(k)_la_mu = T(n-k)_la_mu = T(k)_mu_la = T(k)_la´_mu´ 
and  det( T(k)_la_mu )=0
and T(k)_la_mu  has non-negative integer eigenvalues.
and T(k)_la_mu  has integer eigenvectors.

The Sum (k=1..n-1) T(k)_la_mu  has the same properties as above,  and the eigenvalues equal
(2^k-2) each with multiplicity P(n,k) ; 0<k<=n.


Example 2: 

take n=6, so we make the 11 by 11 matrix of the partitions of n=6, and choose k=1
(so we 'divide' and re-multiply by all partitions of 1 in turn and sum the result to get a matrix element), then T(k=1) is (by revlex ordering) :

T(1)=T(5)

    1    1    0    0    0    0    0    0    0    0    0
    1    2    1    1    0    0    0    0    0    0    0
    0    1    2    1    1    1    0    0    0    0    0
    0    1    1    2    0    1    1    0    0    0    0
    0    0    1    0    1    1    0    0    0    0    0
    0    0    1    1    1    3    1    1    1    0    0
    0    0    0    1    0    1    2    0    1    1    0
    0    0    0    0    0    1    0    1    1    0    0
    0    0    0    0    0    1    1    1    2    1    0
    0    0    0    0    0    0    1    0    1    2    1
    0    0    0    0    0    0    0    0    0    1    1

and T(2)=T(4)

    1    1    1    0    0    0    0    0    0    0    0
    1    3    2    2    1    1    0    0    0    0    0
    1    2    4    2    1    3    1    1    0    0    0
    0    2    2    4    1    3    2    0    1    0    0
    0    1    1    1    2    2    0    0    1    0    0
    0    1    3    3    2    6    3    2    3    1    0
    0    0    1    2    0    3    4    1    2    2    0
    0    0    1    0    0    2    1    2    1    1    0
    0    0    0    1    1    3    2    1    4    2    1
    0    0    0    0    0    1    2    1    2    3    1
    0    0    0    0    0    0    0    0    1    1    1

and T(3)

    1    1    1    0    1    0    0    0    0    0    0
    1    3    3    2    1    2    0    0    0    0    0
    1    3    4    3    2    4    1    1    1    0    0
    0    2    3    5    1    4    3    1    1    0    0
    1    1    2    1    2    2    1    1    1    0    0
    0    2    4    4    2    8    4    2    4    2    0
    0    0    1    3    1    4    5    1    3    2    0
    0    0    1    1    1    2    1    2    2    1    1
    0    0    1    1    1    4    3    2    4    3    1
    0    0    0    0    0    2    2    1    3    3    1
    0    0    0    0    0    0    0    1    1    1    1

and their eigenvalues are
T(1) :: { 6,4,3,2,2,1,1,0,0,0,0}
T(2) :: {15,7,3,3,3,1,1,1,0,0,0}
T(3) :: {20,8,4,2,2,2,0,0,0,0,0}

the sum (k=1..5, T(k) ) is
5     5    3    0    1      0    0    0    0    0    0
5   13    9    8    3      4    0    0    0    0    0
3     9  16    9    6    12    3    3    1    0    0
0     8    9  17    3    12    9    1    3    0    0
1     3    6    3    8      8    1    1    3    0    0
0     4  12  12    8    26  12    8  12    4    0
0     0    3    9    1    12  17    3    9    8    0
0     0    3    1    1      8    3    8    6    3    1
0     0    1    3    3    12    9    6  16    9    3
0     0    0    0    0      4    8    3    9  13    5
0     0    0    0    0      0    0    1    3    5    5

with eigenvalues
{62, 30, 14, 14, 6, 6, 6, 2, 2, 2, 0}

this appears to be  (2^k-2)  =  0,2,6,14,30,62  with multiplicity  P(6,k) = 1,3,3,2,1,1  or (A008284 : “Triangle of partition numbers: T(n,k) = number of partitions of n in which the greatest part is k” )

Question:
it appears that the eigenvalues and their multiplicities of the partial matrices T(k) follow a different set of sequences:
T(1): {0,1,2,3,4,6}   and {4,2,2,1,1,1}
T(2): {0,1,3,7,15}    and {3,3,3,1,1}
T(3): {0,2,4,8,20}    and {5,3,1,1,1}

What is a necessary and sufficient condition for integer matrices to have integer eigenvalues and eigenfunctions? 
Do they ‘count’ something?


Wouter.