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 Î N be a positive
integer. For every permutation p Î Sr the defect set
of p is defined by
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ì
í
î
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k Î { 1 , ¼, r - 1 } | p(k) > p(k + 1) |
ü
ý
þ
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Solomon's descent algebra1
is the linear
K-subspace
Dr of
K [
Sr] which is
generated by all elements
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å
[(p Î Sr) || D(p) = D]
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p , D Í { 1 , ¼, r - 1 } . |
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Dr is multiplicatively closed and, consequently, a subalgebra of
K [Sr].
If q = (q
1 ,
¼q
k)
| = r is a
decomposition2
of r
Î N, then the
standard partition
relative to q is defined to be the k-tuple
Q
q : = (Q
1q ,
¼, Q
kq) of sets
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| Qjq : = { i Î N | q0 + q1 + ¼+ qj - 1 + 1 £ i £ q1 + ¼+ qj } , q0 : = 0 . |
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Now we define the permutation sets
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ì
í
î
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p Î Sr | p |Qjq is increasing for all j Î {1 , ¼, k} |
ü
ý
þ
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ì
í
î
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p Î Sr | p |Qjq has a unique local minimum for all j Î {1 , ¼, k} |
ü
ý
þ
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ì
í
î
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p Î Xq | p(Qjq) = Qjq for all j Î {1 , ¼, k} |
ü
ý
þ
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and the group ring elements
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Xq : = |
å
p Î Sq
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p , wq : = |
å
p Î Xq
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(-1)dq(p) p , wq : = Xq ·wq , |
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where dq(p) denotes the defect
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k
å
i = 1
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|{ j | p(j) > p(j + 1) and j , j+1 Î Qiq }| |
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of a permutation p Î Xq.
The sets of group ring elements
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ì
í
î
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dD | D Í {1,¼,r-1} |
ü
ý
þ
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, |
ì
í
î
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Xq | q | = r |
ü
ý
þ
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ì
í
î
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wq | q | = r |
ü
ý
þ
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are K-bases of Dr. Furthermore, the wq generate
indecomposable right ideals Lq : = wq ·Dr
of Dr and Dr is the direct sum
Dr = Ål |- 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 , ¼, qk) | = r is a decomposition, then q? denotes
the number
q? : = Õi = 1k qi ·Õj = 1r (aj(q)!) where
aj(q) : = |{ i Î {1 , ¼, k} | qi = j }|. Then
is an idempotent generator of Lq.
The following tables contain the multiplicities of the decompositions of the
left ideals
ll : = C [Sr]·nl into minimal left
ideals and the characters of wSr |ll
for all idempotents
nl with l |- r and r = 2 , ¼, 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:
- In a table of character values, m |- r denotes the conjugacy
class Km of Sr on which the given character value is taken.
- 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 [Sr]·nli
(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.
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| class\l | (12) | (2) |
| (12) | 1 | 1 |
| (2) | 1 | -1 |
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| rep.\l | (12) | (2) |
| (12) | 0 | 1 |
| (2) | 1 | 0 |
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Table 2: Characters and multiplicities connected with the
nl of the S3.
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| class\l | (13) | (2 1) | (3) |
| (13) | 1 | 3 | 2 |
| (2 1) | 1 | -1 | 0 |
| (3) | 1 | 0 | -1 |
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| rep.\l | (13) | (2 1) | (3) |
| (13) | 0 | 1 | 0 |
| (2 1) | 0 | 1 | 1 |
| (3) | 1 | 0 | 0 |
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Table 3: Characters connected with the nl of the
S4.
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| 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 |
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Table 4: Multiplicities connected with the nl of the
S4.
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| 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 |
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Table 5: Characters connected with the nl of the
S5.
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| 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 |
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Table 6: Multiplicities connected with the nl of the
S5.
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| 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 |
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Table 7: Characters connected with the nl of the
S6.
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| 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 |
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Table 8: Multiplicities connected with the nl of the
S6.
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| 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 |
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Table 9: Characters connected with the nl of the
S7.
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| 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 |
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Table 10: Multiplicities connected with the nl of the
S7.
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| 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 |
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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 , ¼, qk) of positive integers qj with
q1 + ¼+ qk = r.
File translated from TEX by TTH, version 2.10.
On 17 Jul 1999, 01:40.
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