| eBook ISBN: | 978-1-4704-0292-1 |
| Product Code: | MEMO/148/701.E |
| List Price: | $52.00 |
| MAA Member Price: | $46.80 |
| AMS Member Price: | $31.20 |
| eBook ISBN: | 978-1-4704-0292-1 |
| Product Code: | MEMO/148/701.E |
| List Price: | $52.00 |
| MAA Member Price: | $46.80 |
| AMS Member Price: | $31.20 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 148; 2000; 109 ppMSC: Primary 55; 20; 16; 17
Abstract. We consider functorial decompositions of \(\Omega\Sigma X\) in the case where \(X\) is a \(p\)-torsion suspension. By means of a geometric realization theorem, we show that the problem can be reduced to the one obtained by applying homology: that of finding natural coalgebra decompositions of tensor algebras. We solve the algebraic problem and give properties of the piece \(A^{\mathrm{min}}(V)\) of the decomposition of \(T(V)\) which contains \(V\) itself, including verification of the Cohen conjecture that in characteristic \(p\) the primitives of \(A^{\mathrm{min}}(V)\) are concentrated in degrees of the form \(p^t\). The results tie in with the representation theory of the symmetric group and in particular produce the maximum projective submodule of the important \(S_n\)-module \(\mathrm{Lie}(n)\).
ReadershipGraduate students and research mathematicians interested in topology and representation theory.
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Table of Contents
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Chapters
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1. Introduction
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2. Natural coalgebra transformations of tensor algebras
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3. Geometric realizations and the proof of Theorem 1.3
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4. Existence of minimal natural coalgebra retracts of tensor algebras
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5. Some lemmas on coalgebras
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6. Functorial version of the Poincaré-Birkhoff-Witt theorem
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7. Projective $\mathbf {k}(S_n)$-submodules of $\operatorname {Lie}(n)$
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8. The functor $A^{\mathrm {min}}$ over a field of characteristic $p > 0$
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9. Proof of Theorems 1.1 and 1.6
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10. The functor $L’_n$ and the associated $\mathbf {k}(\Sigma _n)$-module $\operatorname {Lie}’(n)$
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11. Examples
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Abstract. We consider functorial decompositions of \(\Omega\Sigma X\) in the case where \(X\) is a \(p\)-torsion suspension. By means of a geometric realization theorem, we show that the problem can be reduced to the one obtained by applying homology: that of finding natural coalgebra decompositions of tensor algebras. We solve the algebraic problem and give properties of the piece \(A^{\mathrm{min}}(V)\) of the decomposition of \(T(V)\) which contains \(V\) itself, including verification of the Cohen conjecture that in characteristic \(p\) the primitives of \(A^{\mathrm{min}}(V)\) are concentrated in degrees of the form \(p^t\). The results tie in with the representation theory of the symmetric group and in particular produce the maximum projective submodule of the important \(S_n\)-module \(\mathrm{Lie}(n)\).
Graduate students and research mathematicians interested in topology and representation theory.
-
Chapters
-
1. Introduction
-
2. Natural coalgebra transformations of tensor algebras
-
3. Geometric realizations and the proof of Theorem 1.3
-
4. Existence of minimal natural coalgebra retracts of tensor algebras
-
5. Some lemmas on coalgebras
-
6. Functorial version of the Poincaré-Birkhoff-Witt theorem
-
7. Projective $\mathbf {k}(S_n)$-submodules of $\operatorname {Lie}(n)$
-
8. The functor $A^{\mathrm {min}}$ over a field of characteristic $p > 0$
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9. Proof of Theorems 1.1 and 1.6
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10. The functor $L’_n$ and the associated $\mathbf {k}(\Sigma _n)$-module $\operatorname {Lie}’(n)$
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11. Examples
