## Algebra & Number Theory

### Block components of the Lie module for the symmetric group

#### Abstract

Let $F$ be a field of prime characteristic $p$ and let $B$ be a nonprincipal block of the group algebra $FSr$ of the symmetric group $Sr$. The block component $Lie(r)B$ of the Lie module $Lie(r)$ is projective, by a result of Erdmann and Tan, although $Lie(r)$ itself is projective only when $p∤r$. Write $r=pmk$, where $p∤k$, and let $Sk∗$ be the diagonal of a Young subgroup of $Sr$ isomorphic to $Sk×⋯×Sk$. We show that $pmLie(r)B≅(Lie(k)↑Sk∗Sr)B$. Hence we obtain a formula for the multiplicities of the projective indecomposable modules in a direct sum decomposition of $Lie(r)B$. Corresponding results are obtained, when $F$ is infinite, for the $r$-th Lie power $Lr(E)$ of the natural module $E$ for the general linear group $GLn(F)$.

#### Article information

Source
Algebra Number Theory, Volume 6, Number 4 (2012), 781-795.

Dates
Received: 10 March 2011
Revised: 8 June 2011
Accepted: 6 July 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729823

Digital Object Identifier
doi:10.2140/ant.2012.6.781

Mathematical Reviews number (MathSciNet)
MR2966719

Zentralblatt MATH identifier
1247.20011

#### Citation

Bryant, Roger; Erdmann, Karin. Block components of the Lie module for the symmetric group. Algebra Number Theory 6 (2012), no. 4, 781--795. doi:10.2140/ant.2012.6.781. https://projecteuclid.org/euclid.ant/1513729823

#### References

• D. J. Benson, Representations and cohomology, vol. 1: Basic representation theory of finite groups and associative algebras, Cambridge Studies in Advanced Mathematics 30, Cambridge University Press, 1995.
• N. Bourbaki, Groupes et algèbres de Lie: Chapitre II: Algèbres de Lie libres; Chapitre III: Groupes de Lie, Actualités Scientifiques et Industrielles 1349, Hermann, Paris, 1972.
• R. M. Bryant, “Lie powers of infinite-dimensional modules”, Beiträge Algebra Geom. 50:1 (2009), 179–193.
• S. Donkin, “On Schur algebras and related algebras, IV: The blocks of the Schur algebras”, J. Algebra 168:2 (1994), 400–429.
• S. Donkin and K. Erdmann, “Tilting modules, symmetric functions, and the module structure of the free Lie algebra”, J. Algebra 203:1 (1998), 69–90.
• K. Erdmann, “Symmetric groups and quasi-hereditary algebras”, pp. 123–161 in Finite-dimensional algebras and related topics (Ottawa, 1992), edited by V. Dlab and L. L. Scott, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 424, Kluwer, Dordrecht, 1994.
• K. Erdmann and K. M. Tan, “The non-projective part of the Lie module for the symmetric group”, Arch. Math. $($Basel$)$ 96:6 (2011), 513–518.
• J. A. Green, Polynomial representations of ${\rm GL}_{n}$, Lecture Notes in Mathematics 830, Springer-Verlag, Berlin, 1980.
• G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications 16, Addison-Wesley, Reading, MA, 1981.
• K. J. Lim and K. M. Tan, “The Schur functor on tensor powers”, Arch. Math. $($Basel$)$ 98:2 (2012), 99–104.
• H. Nagao and Y. Tsushima, Representations of finite groups, Academic Press, Boston, MA, 1989.