Characters and multiplicities connected with idempotents of Solomon's algebra

B. Fiedler
Alfred-Rosch-Str. 13, D-04249 Leipzig
email: Bernd.Fiedler.RoschStr.Leipzig@t-online.de

April, 1999

The following Tables are an example of calculations by means of my Mathematica package PERMS. Their making was suggested by an inquiry of Th. Bauer, A. J"ollenbeck and M. Schocker (Mathematisches Seminar, Universität Kiel), who were interested in information about a decomposition of the below idempotents nq into pairwise orthogonal, primitive idempotents. As a result of these calculations several routines of PERMS were improved and some new tools were added to PERMS.

The present text about the tables is a part of my postdoctoral thesis. The HTML version of the tables is not legible under the Netscape Communicator 2.x since certain fonts can not be used. If a later version of the Netscape Communicator does not yield a correct result, then modify the entries of your files .Xdefaults or .Xresources by means of the instructions of Ian Hutchinson.

Characters and multiplicities connected with the nl

Let K be a field of characteristic 0 and r N be a positive integer. For every permutation p Sr the defect set of p is defined by
D(p)
: =


k { 1 , , r - 1 }   |  p(k) > p(k + 1)

 .
Solomon's descent algebra1 is the linear K-subspace Dr of K [Sr] which is generated by all elements
dD
: =


[(p Sr) || D(p) = D] 
 p       ,      D { 1 , , r - 1 }  .
Dr is multiplicatively closed and, consequently, a subalgebra of K [Sr].
 
If q = (q1 , qk)  | =  r is a decomposition2 of r N, then the standard partition relative to q is defined to be the k-tuple Qq : = (Q1q , , Qkq) of sets
Qjq : = { i N   |  q0 + q1 + + qj - 1 + 1 i q1 + + qj }    ,    q0 : = 0  .
Now we define the permutation sets
Sq
: =


p Sr   |  p |Qjq  is increasing for all  j {1 , , k}

 ,
Xq
: =


p Sr   |  p |Qjq  has a unique local minimum for all  j {1 , , k}

 ,
Xq
: =


p Xq   |  p(Qjq) = Qjq   for all  j {1 , , k}

and the group ring elements
Xq : =

p Sq 
 p       ,      wq : =

p Xq 
 (-1)dq(p) p       ,      wq : = Xq ·wq  ,
where dq(p) denotes the defect
dq(p)
: =
k

i = 1 
 |{ j   |  p(j) > p(j + 1)   and  j , j+1 Qiq }|
of a permutation p Xq.
 
The sets of group ring elements


dD   |  D {1,,r-1}

      ,      

Xq   |  q  | =  r

      ,      

wq   |  q  | =  r

are K-bases of Dr. Furthermore, the wq generate indecomposable right ideals Lq : = wq ·Dr of Dr and Dr is the direct sum Dr = l |- r Ll of such right ideals Lq. This direct sum runs over exactly those Lq for which the decomposition q  | =  r is a usual partition q = l |- r.
 
If q = (q1 , , qk)  | =  r is a decomposition, then q? denotes the number q? : = i = 1k qi ·j = 1r (aj(q)!) where aj(q) : = |{ i {1 , , k}   |  qi = j }|. Then
nq
: =
1
q?
 wq
(1)
is an idempotent generator of Lq.

The following tables contain the multiplicities of the decompositions of the left ideals ll : = C [Srnl into minimal left ideals and the characters of wSr |ll for all idempotents nl with l  |- r and r = 2 , , 7. (wSr denotes the regular representation of the Sr.) Every column of such a table belongs to an idempotent nl, which is recognizable by the partition l that numbers the column. The partitions m |- r that number the rows of the tables have the following meaning:

  1. In a table of character values, m |- r denotes the conjugacy class Km of Sr on which the given character value is taken.
  2. In a table of multiplicities, m |- r denotes the class of equivalent minimal left ideals to which the given multiplicity is related.

It is remarkable that the Tables of S4 and S7 contain pairs of idempotents nl1 , nl2 for which the representations w|lli (i = 1 , 2) possess the same character such that the left ideals lli = C [Srnli (i = 1 , 2) are equivalent. The partitions li of these idempotents are (2 12)  , (4) in S4 and (3 2 12) , (4 3) in S7.

 

Table 1: Characters and multiplicities connected with the nl of the S2.
class\l (12) (2)
(12) 1 1
(2) 1 -1
rep.\l (12) (2)
(12) 0 1
(2) 1 0

 

Table 2: Characters and multiplicities connected with the nl of the S3.
class\l (13) (2 1) (3)
(13) 1 3 2
(2 1) 1 -1 0
(3) 1 0 -1
rep.\l (13) (2 1) (3)
(13) 0 1 0
(2 1) 0 1 1
(3) 1 0 0

 

Table 3: Characters connected with the nl of the S4.
class\l (14) (2 12) (22) (3 1) (4)
(14) 1 6 3 8 6
(2 12) 1 0 -1 0 0
(22) 1 -2 3 0 -2
(3 1) 1 0 0 -1 0
(4) 1 0 -1 0 0

 

Table 4: Multiplicities connected with the nl of the S4.
rep.\l (14) (2 12) (22) (3 1) (4)
(14) 0 0 1 0 0
(2 12) 0 1 0 1 1
(22) 0 0 1 1 0
(3 1) 0 1 0 1 1
(4) 1 0 0 0 0

 

Table 5: Characters connected with the nl of the S5.
class\l (15) (2 13) (22 1) (3 12) (3 2) (4 1) (5)
(15) 1 10 15 20 20 30 24
(2 13) 1 2 -3 2 -2 0 0
(22 1) 1 -2 3 0 0 -2 0
(3 12) 1 1 0 -1 -1 0 0
(3 2) 1 -1 0 -1 1 0 0
(4 1) 1 0 -1 0 0 0 0
(5) 1 0 0 0 0 0 -1

 

Table 6: Multiplicities connected with the nl of the S5.
rep.\l (15) (2 13) (22 1) (3 12) (3 2) (4 1) (5)
(15) 0 0 1 0 0 0 0
(2 13) 0 0 1 0 1 1 1
(22 1) 0 0 1 1 1 1 1
(3 12) 0 1 0 1 1 2 1
(3 2) 0 0 1 1 1 1 1
(4 1) 0 1 0 1 0 1 1
(5) 1 0 0 0 0 0 0

 

Table 7: Characters connected with the nl of the S6.
class\l (16) (2 14) (22 12) (23) (3 13) (3 2 1) (32) (4 12) (4 2) (5 1) (6)
(16) 1 15 45 15 40 120 40 90 90 144 120
(2 14) 1 5 -3 -3 8 -8 0 6 -6 0 0
(22 12) 1 -1 1 3 0 0 0 -2 -2 0 0
(23) 1 -3 9 -7 0 0 8 -6 6 0 -8
(3 13) 1 3 0 0 1 -3 -2 0 0 0 0
(3 2 1) 1 -1 0 0 -1 1 0 0 0 0 0
(32) 1 0 0 3 -2 0 1 0 0 0 -3
(4 12) 1 1 -1 -1 0 0 0 0 0 0 0
(4 2) 1 -1 -1 1 0 0 0 0 0 0 0
(5 1) 1 0 0 0 0 0 0 0 0 -1 0
(6) 1 0 0 -1 0 0 -1 0 0 0 1

 

Table 8: Multiplicities connected with the nl of the S6.
rep.\l (16) (2 14) (22 12) (23) (3 13) (3 2 1) (32) (4 12) (4 2) (5 1) (6)
(16) 0 0 0 1 0 0 0 0 0 0 0
(2 14) 0 0 1 0 0 1 0 0 1 1 1
(22 12) 0 0 0 1 0 2 0 1 1 2 2
(23) 0 0 1 0 0 1 1 0 1 1 0
(3 13) 0 0 1 0 0 2 1 1 2 2 1
(3 2 1) 0 0 1 0 1 3 1 2 2 3 3
(32) 0 0 0 1 0 1 0 1 0 1 1
(4 12) 0 1 0 0 1 1 0 2 1 2 2
(4 2) 0 0 1 0 1 1 1 1 1 2 1
(5 1) 0 1 0 0 1 0 0 1 0 1 1
(6) 1 0 0 0 0 0 0 0 0 0 0

 

Table 9: Characters connected with the nl of the S7.
class\l (17) (2 15) (22 13) (23 1) (3 14) (3 2 12) (3 22) (32 1) (4 13) (4 2 1) (4 3) (5 12) (5 2) (6 1) (7)
(17) 1 21 105 105 70 420 210 280 210 630 420 504 504 840 720
(2 15) 1 9 5 -15 20 0 -20 0 30 -30 0 24 -24 0 0
(22 13) 1 1 -3 9 2 -4 6 0 -2 -6 -4 0 0 0 0
(23 1) 1 -3 9 -7 0 0 0 8 -6 6 0 0 0 -8 0
(3 14) 1 6 3 0 7 -6 -3 -8 6 0 -6 0 0 0 0
(3 2 12) 1 0 -1 0 -1 0 1 0 0 0 0 0 0 0 0
(3 22) 1 -2 3 0 -1 2 -3 0 -2 0 2 0 0 0 0
(32 1) 1 0 0 3 -2 0 0 1 0 0 0 0 0 -3 0
(4 13) 1 3 -1 -3 2 0 -2 0 0 0 0 0 0 0 0
(4 2 1) 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0
(4 3) 1 0 -1 0 -1 0 1 0 0 0 0 0 0 0 0
(5 12) 1 1 0 0 0 0 0 0 0 0 0 -1 -1 0 0
(5 2) 1 -1 0 0 0 0 0 0 0 0 0 -1 1 0 0
(6 1) 1 0 0 -1 0 0 0 -1 0 0 0 0 0 1 0
(7) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1

 

Table 10: Multiplicities connected with the nl of the S7.
rep.\l (17) (2 15) (22 13) (23 1) (3 14) (3 2 12) (3 22) (32 1) (4 13) (4 2 1) (4 3) (5 12) (5 2) (6 1) (7)
(17) 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
(2 15) 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1
(22 13) 0 0 0 1 0 1 1 0 0 2 1 1 2 3 2
(23 1) 0 0 0 1 0 1 1 1 0 2 1 1 2 2 2
(3 14) 0 0 1 0 0 1 1 1 0 3 1 1 2 2 2
(3 2 12) 0 0 0 1 0 3 2 2 1 5 3 3 4 6 5
(3 22) 0 0 1 0 0 2 1 2 0 3 2 2 2 3 3
(32 1) 0 0 0 1 0 2 1 1 1 2 2 2 2 4 3
(4 13) 0 0 1 0 0 2 0 1 1 3 2 2 2 3 3
(4 2 1) 0 0 1 0 1 3 1 2 2 4 3 4 3 6 5
(4 3) 0 0 0 1 0 1 1 1 1 1 1 2 1 2 2
(5 12) 0 1 0 0 1 1 0 0 2 1 1 2 1 3 2
(5 2) 0 0 1 0 1 1 0 1 1 1 1 2 1 2 2
(6 1) 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1
(7) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

References

[1]
Dieter Blessenohl and Hartmut Laue. Algebraic combinatorics related to the free Lie algebra. In Adalbert Kerber, editor, Actes 29e Séminaire Lotharingien de Combinatoire, Thurnau, September 1992, number 33 in Publ. I.R.M.A. Strasbourg, pages 1 - 21, 7, rue René Descartes, 67084 Strasbourg Cedex, 1993. Institut de Recherche Mathématique Avancée, Université Louis Pasteur et C.N.R.S. (URA 01).
 
[2]
Dieter Blessenohl and Hartmut Laue. The module structure of Solomon's descent algebra. preprint, Mathematisches Seminar, Universität Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, 1994. In: Internet preprint archive of the Lehrstuhl II für Mathematik, Department of Mathematics, University of Bayreuth.
 
[3]
Dieter Blessenohl and Hartmut Laue. On the descending Loewy series of Solomon's descent algebra. preprint, Mathematisches Seminar, Universität Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, 1994. In: Internet preprint archive of the Lehrstuhl II für Mathematik, Department of Mathematics, University of Bayreuth.
 

Footnotes:

1 The above facts about Solomon's algebra can be found in the papers [2,3] and [1] of D. Blessenohl and H. Laue. Note that these papers use the convention p s(i) = s(p(i)) for the multiplication of permutations.

2 A decomposition of a natural number r is a finite sequence q = (q1 , , qk) of positive integers qj with q1 + + qk = r.


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