Ask Your Question
2

Define matrix indexed by partitions

asked 2021-06-29 21:00:56 +0200

mathstudent gravatar image

I want to define the following matrix in sage: $J$ is of size Partitions(n), and $J_{\lambda, \mu} = 1$ if $\lambda'=\mu$ and $0$ else. Can someone please help?

edit retag flag offensive close merge delete

2 Answers

Sort by » oldest newest most voted
1

answered 2021-06-29 21:48:31 +0200

tmonteil gravatar image

updated 2021-06-30 01:56:45 +0200

I am not sure about the notation $\lambda'$ so let me assume that it is the conjugate.

First, to build a matrix, you have to index the partitions, which is easy with the enumerate function:

sage: n = 6
sage: P = Partitions(n)
sage: m = matrix(ZZ, P.cardinality())
sage: list(enumerate(P))
[(0, [6]),
 (1, [5, 1]),
 (2, [4, 2]),
 (3, [4, 1, 1]),
 (4, [3, 3]),
 (5, [3, 2, 1]),
 (6, [3, 1, 1, 1]),
 (7, [2, 2, 2]),
 (8, [2, 2, 1, 1]),
 (9, [2, 1, 1, 1, 1]),
 (10, [1, 1, 1, 1, 1, 1])]

In the other way, given a partition, you need to be able to find its index, which is doable with this dictionary:

sage: part_to_int = {p:i for i,p in enumerate(P)}

Now, you can construct your matrix:

sage: for i,p in enumerate(P):
....:     j = part_to_int[p.conjugate()]
....:     m[i,j] = 1

You have:

sage: m
[0 0 0 0 0 0 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0]
[1 0 0 0 0 0 0 0 0 0 0]

sage: m^2
[1 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 0 1]

If $\lambda'$ does not denote the conjugate, you can adapt by replacing p.conjugate() with your favorite function.

edit flag offensive delete link more

Comments

This was exactly what I was looking for. Thank you so much.

mathstudent gravatar imagemathstudent ( 2021-06-29 22:30:34 +0200 )edit
1

answered 2021-06-29 23:10:28 +0200

Nicolas M. Thiéry gravatar image

updated 2021-06-30 01:06:36 +0200

slelievre gravatar image

Potential alternatives:

edit flag offensive delete link more

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

1 follower

Stats

Asked: 2021-06-29 21:00:56 +0200

Seen: 51 times

Last updated: Jun 30