Đến nội dung

Hình ảnh

Conjugacy class

- - - - -

  • Please log in to reply
Chủ đề này có 5 trả lời

#1
RongChoi

RongChoi

    Thượng sĩ

  • Founder
  • 215 Bài viết
Yet another small problem for this new and interesting topic :)

Let http://dientuvietnam...mimetex.cgi?S_n be the symmetric group of degree http://dientuvietnam...n/mimetex.cgi?n and let
http://dientuvietnam.net/cgi-bin/mimetex.cgi?K_n
ii) Prove that, for any http://dientuvietnam.net/cgi-bin/mimetex.cgi?\pi is equal to http://dientuvietnam...mimetex.cgi?K_n
iii) http://dientuvietnam...mimetex.cgi?K_n

#2
noproof

noproof

    Trung sĩ

  • Thành viên
  • 104 Bài viết
I have solved this "small" problem in the case http://dientuvietnam...n/mimetex.cgi?n is odd :leq. In this case, http://dientuvietnam...mimetex.cgi?K_n is empty and all statements are followed.

(I think Rong choi had a typing mistake in the formula defining http://dientuvietnam...imetex.cgi?K_n:

.)

#3
madness

madness

    Trung sĩ

  • Thành viên
  • 137 Bài viết
Can I try? :leq

Prop. 1: an element x in S_n is an involution, i.e. x^2=id, if and only if x is a product of disjoint transpositions.

Prop. 2: 2 elements in S_n are in the same conjugacy class if and only if they have the same cycle structure.

For the case when n is even (n=2m):

(i) The elements in K_n are precisely the elements in S_n which are the products of m disjoint transpositions. A combinatorial counting will give us the size of K_n as follows: |K_n| = (2m)! / (m!*2^m).

(ii) Straightforward from Prop. 2.

(iii) I don't understand what "uniformly distributed" means in this case, so I guess it must be one of the following meanings:

a. "uniformly distributed" means: txt^(-1) runs through K_n when x runs through K_n. This can be seen from the fact that o(txt^(-1))=2 and txt^(-1) (i) <> i for all i in {1,2,...,n}.

b. "uniformly distributed" means: txt^(-1) runs through K_n "uniformly" when t runs through S_n. This can be seen from the fact that txt^(-1)=axa^(-1) iff t and a make the same (left) cosets of C(x) in S_n (the centralizer of x in S_n), i.e. tC(x)=aC(x). The number of such cosets is exactly the cardinality of the conjugacy class of x, which is |K_n|. Thus txt^(-1) runs through K_n "uniformly" (i.e. runs through K_n for |C(x)| times) when t runs through S_n.

#4
RongChoi

RongChoi

    Thượng sĩ

  • Founder
  • 215 Bài viết
@ noproof: thanks to fix the typo.
@ madness:
i) and ii) : your proposed solution is very clear :leq
iii) : ìweak (but general) uniformed distribution” means that, .
ìstrong uniformed distribution” combines your a) and b) definitions.
Your explanation in b) seems correct but I have not completely seen it yet… :leq

#5
madness

madness

    Trung sĩ

  • Thành viên
  • 137 Bài viết
From your interpretation of "weak uniformly distributed", this can be seen from the fact stated in a) alone.

My explanation in b) is not clear :P Let me try to make it clearer.

b. "uniformly distributed" means: txt^(-1) runs through K_n "uniformly" when x \in S_n is fixed t runs through S_n. Now we try to understand the behavior of txt^(-1) when t runs and x is fixed.

Note that txt^(-1)=axa^(-1) iff a^(-1)t.x.t^(-1)a=x iff a^(-1)t.x=x.a^(-1)t iff a^(-1)t \in C(x) (the centralizer of x in S_n), i.e. tC(x)=aC(x).

The number of left cosets of C(x) in S_n is [S_n:C(x)], this is exactly the cardinality of the conjugacy class of x, which is |K_n|. Thus txt^(-1) runs through K_n "uniformly" (i.e. runs through K_n for |C(x)| times) when t runs through S_n.

:beer

#6
5W-H

5W-H

    Lính mới

  • Thành viên
  • 5 Bài viết
Very well! :clap




1 người đang xem chủ đề

0 thành viên, 1 khách, 0 thành viên ẩn danh