#### Algebraic and Geometric Topology 2 (2002),
paper no. 27, pages 563-590.

## Linking first occurrence polynomials over F_p by Steenrod operations

### Pham Anh Minh, Grant Walker

**Abstract**.
This paper provides analogues of the results of [G.Walker and
R.M.W. Wood, Linking first occurrence polynomials over F_2 by Steenrod
operations, J. Algebra 246 (2001), 739--760] for odd primes p. It is
proved that for certain irreducible representations L(\lambda ) of the
full matrix semigroup M_n(F_p), the first occurrence of L(\lambda ) as
a composition factor in the polynomial algebra P=F_p[x_1,...,x_n] is
linked by a Steenrod operation to the first occurrence of L(\lambda )
as a submodule in P. This operation is given explicitly as the image
of an admissible monomial in the Steenrod algebra A_p under the
canonical anti-automorphism \chi . The first occurrences of both kinds
are also linked to higher degree occurrences of L(\lambda ) by
elements of the Milnor basis of A_p.
**Keywords**.
Steenrod algebra, anti-automorphism, p-truncated polynomial algebra T, T-regular partition/representation

**AMS subject classification**.
Primary: 55S10.
Secondary: 20C20.

**DOI:** 10.2140/agt.2002.2.563

**E-print:** `arXiv:math.AT/0207213`

Submitted: 24 January 2002.
Accepted: 10 July 2002.
Published: 20 July 2002.

Notes on file formats
Pham Anh Minh, Grant Walker

Department of Mathematics, College of Sciences

University of Hue, Dai hoc Khoa hoc, Hue, Vietnam

and

Department of Mathematics, University of Manchester

Oxford Road, Manchester M13 9PL, U.K.

Email: paminh@dng.vnn.vn, grant@ma.man.ac.uk

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