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## Problems in the Steenrod algebra (1998)

Venue: | Bull. London Math. Soc |

Citations: | 30 - 1 self |

### Citations

630 |
The representation theory of the symmetric group
- James, Kerber
- 1981
(Show Context)
Citation Context ... reference for symmetric functions is Macdonald's book [121], and the representation theory of symmetric groups and general linear groups over finite fields is treated in the book of James and Kerber =-=[90]-=-. The differential operator approach to the Steenrod algebra and the Landweber-Novikov algebra touches on certain combinatorial material which can be found in standard texts such as Aigner [13], Camer... |

582 |
Symmetric Functions and Hall Polynomials, second edition
- Macdonald
- 1995
(Show Context)
Citation Context ...ative chapters 3son the Steenrod algebra, and likewise the books of Adem and Milgram [12] and Benson [23] on the cohomology of groups. A standard reference for symmetric functions is Macdonald's book =-=[121]-=-, and the representation theory of symmetric groups and general linear groups over finite fields is treated in the book of James and Kerber [90]. The differential operator approach to the Steenrod alg... |

393 |
Combinatorial enumeration
- Goulden, Jackson
- 1983
(Show Context)
Citation Context ...e Steenrod algebra and the Landweber-Novikov algebra touches on certain combinatorial material which can be found in standard texts such as Aigner [13], Cameron [37], Comtet [47], Goulden and Jackson =-=[77]-=- and Henrici [80], as well as the classic text of MacMahon [122]. Recent work on the Steenrod algebra and the Landweber-Novikov algebra in relation to combinatorics can be found in [163, 164, 165, 169... |

343 |
Noncommutative noetherian rings
- McConnell, Robson
- 1987
(Show Context)
Citation Context ...lgebra impinges on areas of mathematics concerning D-modules, Lie algebras, deformation of Hopf algebras, quantum groups and the Weyl algebra. Relevant information can be found in a number of sources =-=[1, 27, 48, 76, 97, 124, 125, 127, 129, 144, 194]-=-. The present article expands on material found in [221] but stems originally from a talk entitled "Facts and Fancies in the Steenrod Algebra" delivered during the topology conference at G"ottingen in... |

219 |
Stable homotopy and generalised homology
- Adams
- 1974
(Show Context)
Citation Context ...motopy theory. The integral approach to Steenrod squares is closely related to operations in complex bordism theory, as Landweber noted in his original paper [111]. The Chicago lecture notes of Adams =-=[4]-=- describe Novikov's work on this subject, and a general survey of bordism theory can be found in Stong [190]. More recent references are Ravenel [162] and Kochman [101]. Ever since its inception, the ... |

173 |
Quantum groups. Graduate Texts in Mathematics, 155
- Kassel
- 1995
(Show Context)
Citation Context ...lgebra impinges on areas of mathematics concerning D-modules, Lie algebras, deformation of Hopf algebras, quantum groups and the Weyl algebra. Relevant information can be found in a number of sources =-=[1, 27, 48, 76, 97, 124, 125, 127, 129, 144, 194]-=-. The present article expands on material found in [221] but stems originally from a talk entitled "Facts and Fancies in the Steenrod Algebra" delivered during the topology conference at G"ottingen in... |

151 |
Foundations of quantum group theory (Cambridge
- Majid
- 1995
(Show Context)
Citation Context ...lgebra impinges on areas of mathematics concerning D-modules, Lie algebras, deformation of Hopf algebras, quantum groups and the Weyl algebra. Relevant information can be found in a number of sources =-=[1, 27, 48, 76, 97, 124, 125, 127, 129, 144, 194]-=-. The present article expands on material found in [221] but stems originally from a talk entitled "Facts and Fancies in the Steenrod Algebra" delivered during the topology conference at G"ottingen in... |

134 |
An introduction to noncommutative noetherian rings
- Goodearl, Warfield
- 1989
(Show Context)
Citation Context |

118 | On the non-existence of elements of Hopf invariant one
- Adams
- 1960
(Show Context)
Citation Context ...bra by Sq0 and the Sq2 k for k * 0. In the decade 1950-1960, the Steenrod algebra became one of the most powerful tools in algebraic topology. For example, Adams solved the Hopf invariant one problem =-=[3]-=-, thereby proving that non-singular bilinear maps Rn \ThetasRn ! Rn exist only for dimensions n = 1; 2; 4; 8, where they are realised by real, complex, quaternionic and Cayley multiplication. The fact... |

85 |
Representations and cohomology II: Cohomology of groups and modules
- Benson
- 1991
(Show Context)
Citation Context ...heory and invariant theory [188]. The book on invariant theory by Larry Smith [183] contains informative chapters 3son the Steenrod algebra, and likewise the books of Adem and Milgram [12] and Benson =-=[23]-=- on the cohomology of groups. A standard reference for symmetric functions is Macdonald's book [121], and the representation theory of symmetric groups and general linear groups over finite fields is ... |

64 |
History of Algebraic and Differential Topology,
- Dieudonne
- 1989
(Show Context)
Citation Context ...od algebra [187] at various primes, and the contributions of the early pioneers to the development of the Steenrod algebra are expounded in Dieudonn'e's history of algebraic and differential topology =-=[59]-=-. The Steenrod algebra is a graded Hopf algebra, for which the papers of Milnor [131] and Milnor and Moore [133] are standard references. The differential operator approach to the Steenrod algebra imp... |

63 |
A fundamental system of invariants of the general modular linear group with a solution of the form problem
- Dickson
- 1911
(Show Context)
Citation Context ...currences of the trivial representation in the polynomial algebra can be identified with the ring of invariants of the general linear group GL(n; F2). This is known classically as the Dickson algebra =-=[58, 123, 205, 214]-=-, and is a polynomial subring of W(n) \OmegasF2 generated by the Dickson invariants. The action of the Steenrod squares on the Dickson invariants and, more generally, invariants of parabolic subgroups... |

58 |
Algebraic D-modules
- Borel
- 1987
(Show Context)
Citation Context |

40 |
Cohomology of finite groups, Grundlehren der Mathematischen Wissenschaften 309
- Adem, Milgram
- 1994
(Show Context)
Citation Context ...oup cohomology theory and invariant theory [188]. The book on invariant theory by Larry Smith [183] contains informative chapters 3son the Steenrod algebra, and likewise the books of Adem and Milgram =-=[12]-=- and Benson [23] on the cohomology of groups. A standard reference for symmetric functions is Macdonald's book [121], and the representation theory of symmetric groups and general linear groups over f... |

38 |
Hopf algebras of symmetric functions and class functions
- Geissinger
- 1977
(Show Context)
Citation Context ...the algebraic Thom map. Example 3.1 OE(D(K)) = m(K); OE(Dk) = pk; OE(SQr) = er; OE ` D ffir1 r! ' = hr: We recall that the Dk are the primitives in the Hopf algebra D. In \Lambdasthere is a coproduct =-=[71, 121, 208]-=- which makes \Lambdasinto a Hopf algebra with respect to the dot product. The primitives are the pk. We noted that OE(Dk) = pk, and it can be checked that the algebraic Thom map is a coalgebra map. He... |

31 |
Generic representations of the finite general linear groups and the Steenrod algebra.
- Kuhn
- 1994
(Show Context)
Citation Context ...ed more extensively in Li's work [116]. The global approach is now usually adopted in setting up the Steenrod algebra in an algebraic context. For example, in Larry Smith's book [183] and Kuhn's work =-=[107, 108, 109]-=-, the rules in Lemma 1.3 are extended to cover finite fields through generating functions for total operations. Let Fq denote the Galois field, where q is a power of the prime p. The Steenrod reduced ... |

27 |
Finite H-spaces and algebras over the Steenrod algebra
- Adams, Wilkerson
- 1980
(Show Context)
Citation Context ...e Dickson invariants. The action of the Steenrod squares on the Dickson invariants and, more generally, invariants of parabolic subgroups of general linear groups, is worked out in a number of places =-=[9, 81, 83, 99, 146, 193, 214]-=-, and global product formulae are produced in [116]. The restricted hit problem for the Dickson algebra is solved for a small number of variables in [72, 84, 87, 88], but the general problem appears d... |

26 |
The iteration of Steenrod squares in algebraic topology
- Adem
- 1952
(Show Context)
Citation Context ...mposition, subject to relations which vanish on the cohomology of all spaces X. Wu [224, 222] conjectured certain relations among the squaring operations, which were proved by Adem. Theorem 1.2 (Adem =-=[10, 11]-=-) All relations in the Steenrod algebra are generated by the set of Adem relations SqiSqj = X 0^k^[i=2]sj \Gammask \Gammas1 i \Gammas2k !Sq i+j\Gamma kSqk for 0 ! i ! 2j, where [i=2] denotes the great... |

25 |
Combinatorial Theory.” Grundlehren der mathematischen Wissenschaften
- Aigner
- 1997
(Show Context)
Citation Context ...Kerber [90]. The differential operator approach to the Steenrod algebra and the Landweber-Novikov algebra touches on certain combinatorial material which can be found in standard texts such as Aigner =-=[13]-=-, Cameron [37], Comtet [47], Goulden and Jackson [77] and Henrici [80], as well as the classic text of MacMahon [122]. Recent work on the Steenrod algebra and the Landweber-Novikov algebra in relation... |

21 |
stable homotopy and the Adams spectral sequences
- Kochman, Bordism
- 1996
(Show Context)
Citation Context ...he Chicago lecture notes of Adams [4] describe Novikov's work on this subject, and a general survey of bordism theory can be found in Stong [190]. More recent references are Ravenel [162] and Kochman =-=[101]-=-. Ever since its inception, the Steenrod algebra has been allied to group cohomology theory and invariant theory [188]. The book on invariant theory by Larry Smith [183] contains informative chapters ... |

20 |
Sur la structure des A-modules instables injectifs, Topology 28
- Lannes, Schwartz
- 1989
(Show Context)
Citation Context ...eyl modules for the general linear groups. Information can be found in [69] for the relationship between the first occurrence problems and Lannes' theory of unstable modules over the Steenrod algebra =-=[107, 108, 109, 112, 113]-=-. We can ask for an analogue, for the differential operator algebra, of the AdamsGunawardena-Miller result [7] which states that all grade-preserving linear transformations of the polynomial algebra W... |

19 |
Lie Algebras,” Interscience Tracts
- Jacobson
- 1962
(Show Context)
Citation Context ...ll gradings. A rational basis for N , which is the same as D and U over Q by Theorem 2.5, is given by the classical Poincar'eBirkhoff-Witt theorem in the universal enveloping algebra of a Lie algebra =-=[89]-=-. The set of composites Dffir1k1 ffi Dffir2k2 ffi \Deltas\Deltas\Deltasffi Dffiraka , where k1 ? k2 ? : : : ? ka, forms an additive basis of U . It is not immediately clear that a similar statement is... |

18 |
Über die Steenrodschen Kohomologieoperationen
- Dold
- 1961
(Show Context)
Citation Context ...relates to Smith theory [29, 200, 225], double point sets in bordism with singularities, and SpanierWhitehead duality [130]. Dold sets up the Steenrod algebra for cohomology of a topological space in =-=[60]-=-. Quillen [161] uses Steenrod operations in bordism, further developed by Tom Dieck [202], in connection with formal groups and the Lazard ring. In [25], Bisson and Joyal treat similar topics in terms... |

17 |
Cobordism operations and Hopf algebras
- Landweber
- 1983
(Show Context)
Citation Context ... in the general context of homology and homotopy theory. The integral approach to Steenrod squares is closely related to operations in complex bordism theory, as Landweber noted in his original paper =-=[111]-=-. The Chicago lecture notes of Adams [4] describe Novikov's work on this subject, and a general survey of bordism theory can be found in Stong [190]. More recent references are Ravenel [162] and Kochm... |

16 | Codimension one immersions and the Kervaire invariant one problem
- Eccles
- 1981
(Show Context)
Citation Context ...olynomials divisible by x1 \Deltas\Deltas\Deltasxn. Closed formulae for these actions have been of interest to algebraic and differential topologists, for example in the immersion theory of manifolds =-=[64, 65, 66, 68]-=-. Classically, the Wu formulae [134, 223] answer the question of how the Steenrod squares act on the elementary symmetric polynomials in \Lambda (n) \OmegasF2: Sqk(wm) = wkwm + `k \Gammasm1 'wk\Gamma ... |

15 |
The relations in Steenrod powers of cohomology classes’, Algebraic geometry and topology
- Adem
- 1957
(Show Context)
Citation Context ...mposition, subject to relations which vanish on the cohomology of all spaces X. Wu [224, 222] conjectured certain relations among the squaring operations, which were proved by Adem. Theorem 1.2 (Adem =-=[10, 11]-=-) All relations in the Steenrod algebra are generated by the set of Adem relations SqiSqj = X 0^k^[i=2]sj \Gammask \Gammas1 i \Gammas2k !Sq i+j\Gamma kSqk for 0 ! i ! 2j, where [i=2] denotes the great... |

13 |
Modular representations on the homology of powers of real projective space, Algebraic Topology: Oaxtepec
- Boardman
- 1991
(Show Context)
Citation Context ...lem asks for a description of C(n) 58sas an M (n; F2)-module. Information in the three-variable case can be found in [42, 50, 91], and progress on the general problem can be traced through the papers =-=[15, 26, 50, 178, 179, 182, 218, 219]-=-. Problems still exist in four or more variables. The following results illustrate some of the progress. Recall the numerical function _(d) as in Definition 4.4 as the least number k for which it is p... |

13 |
Characteristic numbers of immersions and self-intersection manifolds’, Topology with applications
- Eccles
- 1993
(Show Context)
Citation Context ...olynomials divisible by x1 \Deltas\Deltas\Deltasxn. Closed formulae for these actions have been of interest to algebraic and differential topologists, for example in the immersion theory of manifolds =-=[64, 65, 66, 68]-=-. Classically, the Wu formulae [134, 223] answer the question of how the Steenrod squares act on the elementary symmetric polynomials in \Lambda (n) \OmegasF2: Sqk(wm) = wkwm + `k \Gammasm1 'wk\Gamma ... |

12 |
On the Adem relations’, Topology 21
- Bullett, Macdonald
- 1982
(Show Context)
Citation Context ...s in these areas can be traced through the work of Crabb, Crossley and Hubbuck [49, 50, 51, 52, 53], Ken Monks [139, 138, 142, 143, 140], and Bill Singer and Judith Silverman [174, 175, 178, 182]. In =-=[36]-=-, Bullett-Macdonald devise a method for generating the Adem relations by equating coefficients in certain products of formal power series. This global approach to the study of squaring operations via ... |

12 |
The composition factors of Fp[x1; x2; x3] as a GL(3; p)-module
- Doty, Walker
- 1992
(Show Context)
Citation Context ..., the first occurrence problem of an irreducible as a submodule has been solved by Schwartz [170] and Tri [204, 203]. Information on the general problem in the context of Weyl modules can be found in =-=[61]-=-. Theorem 7.14 (Schwartz [170]) The first occurrence of the simple M (n; F2)- module corresponding to the column 2-regular partition * as a submodule in Wd(n) \OmegasF2 is for d = X j*1 * j2j\Gamma 1:... |

11 |
The Landweber{Novikov algebra and formal vector on the line’, transl. from Funktsional. Anal. i Prilozhen
- Buhstaber, Shokurov
- 1978
(Show Context)
Citation Context ...iables xi [5, 215]. For interpretations of Landweber-Novikov operations as differential operators in the context of conformal field theory, quantum groups and diffeomorphisms of the line, we refer to =-=[33, 34, 35, 98, 150]-=-. It would also be interesting to know if the complex bordism of a compact Hausdorff topological space X could be defined in an algebraic manner, perhaps in terms of the algebra of complex-valued func... |

11 |
Sur l’iteration des operations de
- Cartan
- 1955
(Show Context)
Citation Context ... grading and never increase length of monomials when the re-write rules are applied to SqiSqj for 0 ! i ! 2j. In his work on the cohomology of Eilenberg-MacLane spaces for the group of order 2, Serre =-=[171, 44]-=- gave a method of deriving the Adem relations in terms of a faithful representation of A on the cohomology of the infinite product of infinite real projective spaces. Let W = Z[x1; : : : ; xn; : : :] ... |

11 |
Truncated symmetric powers and modular representations of GLn
- Doty, Walker
- 1996
(Show Context)
Citation Context ...se known as the classifying space BT n of the n-torus [39, 67, 79, 135, 136, 216, 217]. Little is known about the odd prime analogue of the first occurrence problem, although a few cases are resolved =-=[40, 41, 62, 216]-=-. The first occurrence of a simple module as a submodule is also an interesting question, especially in conjunction with the problem of linking it to the first occurrence as composition factor via Ste... |

10 |
Galois Theory. Notre Dame Mathematical Lectures
- Artin
- 1942
(Show Context)
Citation Context ...x21x22 = \Gamma e22 \Gammase2x21 \Gammase2x22 + (e1e2 \Gammase3)x1 + (e1e2 \Gammase3)x2; where the ei are the elementary symmetric functions. The general case of n variables is worked out by Artin in =-=[18]-=-. The equivalence classes of the n! monomials xi11 xi22 \Deltas\Deltas\Deltasxin \Gamma 1n \Gamma 1 , where 0 ^ ir ^ r, form the Artinbasis of the regular representation C of the symmetric group. All ... |

9 |
The antiautomorphism of the Steenrod algebra
- Davis
- 1974
(Show Context)
Citation Context ...the Steenrod algebra on the cohomology of projective spaces has sparked off a general interest in hit problems. Questions about excess and conjugation in A have been investigated notably by Don Davis =-=[55]-=- and others [20, 21, 70, 103, 116, 191]. Recent advances in these areas can be traced through the work of Crabb, Crossley and Hubbuck [49, 50, 51, 52, 53], Ken Monks [139, 138, 142, 143, 140], and Bil... |

9 |
A-generators for the Dickson algebra
- Hu’ng, Peterson
- 1995
(Show Context)
Citation Context ...n a number of places [9, 81, 83, 99, 146, 193, 214], and global product formulae are produced in [116]. The restricted hit problem for the Dickson algebra is solved for a small number of variables in =-=[72, 84, 87, 88]-=-, but the general problem appears difficult. Hung conjectures that all elements in the positively graded Dickson algebra are hit in W \OmegasF2 if the number of variables is at least three [84, 83, 86... |

9 |
Products of projective spaces as Steenrod modules
- Kameko
- 1990
(Show Context)
Citation Context ... M (n; F2) over the natural field F2 [217]. The equivariant version of the hit problem asks for a description of C(n) 58sas an M (n; F2)-module. Information in the three-variable case can be found in =-=[42, 50, 91]-=-, and progress on the general problem can be traced through the papers [15, 26, 50, 178, 179, 182, 218, 219]. Problems still exist in four or more variables. The following results illustrate some of t... |

8 |
Subalgebras of the Steenrod algebra and the action of matrices on truncated polynomial algebras
- Carlisle, Kuhn
- 1989
(Show Context)
Citation Context ... groups [121]. The transposed matrix corresponds to the conjugate partition *0 of j*j, where *0i is the number of rows k such that *k * i. If * is column 2-regular, then *0 is strictly decreasing. In =-=[40]-=-, a partition with strictly decreasing parts is referred to simply as 2-regular. The first value of the degree d for which a simple module ae occurs as a composition factor in Wd(n) \OmegasF2 gives th... |

8 |
Applied and computational analysis
- Henrici
- 1988
(Show Context)
Citation Context ...a and the Landweber-Novikov algebra touches on certain combinatorial material which can be found in standard texts such as Aigner [13], Cameron [37], Comtet [47], Goulden and Jackson [77] and Henrici =-=[80]-=-, as well as the classic text of MacMahon [122]. Recent work on the Steenrod algebra and the Landweber-Novikov algebra in relation to combinatorics can be found in [163, 164, 165, 169]. Steenrod's ori... |

7 | Representations of the homology of BV and the Steenrod algebra II’, Algebraic topology: new trends in localization and periodicity (San Feliu de Guixols
- Crabb, Hubbuck
- 1994
(Show Context)
Citation Context ...lgebra acts by Kronecker duality. This action is exploited, for example, in studying the splitting of classifying spaces of general linear groups over finite fields [216], and work on the hit problem =-=[15, 16, 50]-=-. An observation which is sometimes useful in dealing with the action of the Steenrod algebra on W \OmegasF2 is the commutativity with partial differentiation. Lemma 2.21 Sqn @@x i = @ @xi Sq n: This ... |

7 |
Poincare series for the occurrence of certain modular representations of GL(n; p
- Carlisle, Walker
- 1989
(Show Context)
Citation Context ...se known as the classifying space BT n of the n-torus [39, 67, 79, 135, 136, 216, 217]. Little is known about the odd prime analogue of the first occurrence problem, although a few cases are resolved =-=[40, 41, 62, 216]-=-. The first occurrence of a simple module as a submodule is also an interesting question, especially in conjunction with the problem of linking it to the first occurrence as composition factor via Ste... |

7 |
The boundedness conjecture for the action of the Steenrod algebra on polynomials
- Carlisle, Wood
- 1992
(Show Context)
Citation Context ... M (n; F2) over the natural field F2 [217]. The equivariant version of the hit problem asks for a description of C(n) 58sas an M (n; F2)-module. Information in the three-variable case can be found in =-=[42, 50, 91]-=-, and progress on the general problem can be traced through the papers [15, 26, 50, 178, 179, 182, 218, 219]. Problems still exist in four or more variables. The following results illustrate some of t... |

7 |
and Dyer{Lashof operations on
- Lance, ‘Steenrod
- 1967
(Show Context)
Citation Context ...n) \OmegasF2: Sqk(wm) = wkwm + `k \Gammasm1 'wk\Gamma 1wm+1 + \Deltas\Deltas\Deltas+ `k \Gammasmk 'wm+k; where negative binomial coefficients are reduced modulo 2. Work on Wu-type formulae appears in =-=[6, 28, 30, 31, 82, 110, 159, 172, 192]-=-. Recently, Lenart [114] has produced integral Wu formulae using differential operators in conjunction with Schur functions, Schubert varieties and the Hammond operators of classical combinatorics. 3.... |

6 |
On the anti-automorphism of the Steenrod algebra
- Barratt, Miller
- 1981
(Show Context)
Citation Context ...ebra on the cohomology of projective spaces has sparked off a general interest in hit problems. Questions about excess and conjugation in A have been investigated notably by Don Davis [55] and others =-=[20, 21, 70, 103, 116, 191]-=-. Recent advances in these areas can be traced through the work of Crabb, Crossley and Hubbuck [49, 50, 51, 52, 53], Ken Monks [139, 138, 142, 143, 140], and Bill Singer and Judith Silverman [174, 175... |

6 |
On the composition of polynomials of the form z2+ cn
- Büger
- 1998
(Show Context)
Citation Context ...ues of ti near to 1, the bound set B(p) of this sequence provides a deformation of the filled-in Julia set of z + z2. It is interesting to study the geometry of B(p) in the spirit of complex dynamics =-=[32]-=-. Of course, the construction of iterated composition may be made in the formal sense on any sequence of formal sequences t1; t2; \Deltas\Deltas\Deltastn; \Deltas\Deltas\Delta . The result is expresse... |

6 |
Reduced unstable A-modules and the modular representation theory of the symmetric groups
- Franjou, Schwartz
- 1990
(Show Context)
Citation Context ...occurrences of \Sigma n modules in the polynomial algebra. James and Kerber [90] treat the theory of modular Specht modules and Weyl modules for the general linear groups. Information can be found in =-=[69]-=- for the relationship between the first occurrence problems and Lannes' theory of unstable modules over the Steenrod algebra [107, 108, 109, 112, 113]. We can ask for an analogue, for the differential... |

6 |
The action of the Steenrod squares on the modular invariants of linear groups
- Hu’ng
- 1991
(Show Context)
Citation Context ...e Dickson invariants. The action of the Steenrod squares on the Dickson invariants and, more generally, invariants of parabolic subgroups of general linear groups, is worked out in a number of places =-=[9, 81, 83, 99, 146, 193, 214]-=-, and global product formulae are produced in [116]. The restricted hit problem for the Dickson algebra is solved for a small number of variables in [72, 84, 87, 88], but the general problem appears d... |

6 | Spherical classes and the Dickson algebra
- ng, Peterson
- 1998
(Show Context)
Citation Context ...n a number of places [9, 81, 83, 99, 146, 193, 214], and global product formulae are produced in [116]. The restricted hit problem for the Dickson algebra is solved for a small number of variables in =-=[72, 84, 87, 88]-=-, but the general problem appears difficult. Hung conjectures that all elements in the positively graded Dickson algebra are hit in W \OmegasF2 if the number of variables is at least three [84, 83, 86... |

6 |
On excess in the Milnor basis
- Kraines
- 1971
(Show Context)
Citation Context ...ebra on the cohomology of projective spaces has sparked off a general interest in hit problems. Questions about excess and conjugation in A have been investigated notably by Don Davis [55] and others =-=[20, 21, 70, 103, 116, 191]-=-. Recent advances in these areas can be traced through the work of Crabb, Crossley and Hubbuck [49, 50, 51, 52, 53], Ken Monks [139, 138, 142, 143, 140], and Bill Singer and Judith Silverman [174, 175... |

5 |
Sub-Hopf algebras of the Steenrod algebra
- Adams, Margolis
- 1974
(Show Context)
Citation Context ...operators [221]. 1.3 Remarks A steady stream of work on the internal structure of the Steenrod algebra and its action on polynomials has continued to the present time. For example, Adams and Margolis =-=[8]-=- classify Hopf subalgebras of A. Frank Peterson's work [157, 158, 156] on the action of the Steenrod algebra on the cohomology of projective spaces has sparked off a general interest in hit problems. ... |

5 | On conjugation invariants in the dual Steenrod algebra
- MD, Whitehouse
(Show Context)
Citation Context ...lop explicit formulae in the differential operator algebra for products of the form D(K) ffi O/(D(L)). Problem 4.26 Describe the fixed point set of conjugation O/ffi in A, D and related Hopf algebras =-=[54]-=-. 5 The stripping technique In theory, any relation E = 0 in the Steenrod algebra can be detected by evaluating E on the product of variables x1 \Deltas\Deltas\Deltasxn if the grading of E does not ex... |

4 |
Groups of formal diffeomorphisms of the superline, generating functions for sequences of polynomials, and functional equations.
- Bukhshtaber, Kholodov
- 1990
(Show Context)
Citation Context ...iables xi [5, 215]. For interpretations of Landweber-Novikov operations as differential operators in the context of conformal field theory, quantum groups and diffeomorphisms of the line, we refer to =-=[33, 34, 35, 98, 150]-=-. It would also be interesting to know if the complex bordism of a compact Hausdorff topological space X could be defined in an algebraic manner, perhaps in terms of the algebra of complex-valued func... |

4 |
Excess and conjugation in the Steenrod algebra
- Gallant
- 1979
(Show Context)
Citation Context ...ebra on the cohomology of projective spaces has sparked off a general interest in hit problems. Questions about excess and conjugation in A have been investigated notably by Don Davis [55] and others =-=[20, 21, 70, 103, 116, 191]-=-. Recent advances in these areas can be traced through the work of Crabb, Crossley and Hubbuck [49, 50, 51, 52, 53], Ken Monks [139, 138, 142, 143, 140], and Bill Singer and Judith Silverman [174, 175... |

4 |
Computing bases for rings of permutation invariant polynomials
- Göbel
- 1995
(Show Context)
Citation Context ...variants, over the integers, of a finite permutation group is a module over D. It would be interesting to investigate this module structure in the case of modular rings of invariants of cyclic groups =-=[75]-=-. We refer to Larry Smith's book [183] for general background on invariant theory, and his recent survey article [184] for up to date information. 7.5 Problems Problem 7.15 Solve the hit problem for t... |

4 |
The action of the mod p Steenrod operations on the modular invariants of linear groups
- Hung
- 1995
(Show Context)
Citation Context ...n a number of places [9, 81, 83, 99, 146, 193, 214], and global product formulae are produced in [116]. The restricted hit problem for the Dickson algebra is solved for a small number of variables in =-=[72, 84, 87, 88]-=-, but the general problem appears difficult. Hung conjectures that all elements in the positively graded Dickson algebra are hit in W \OmegasF2 if the number of variables is at least three [84, 83, 86... |

4 | The combinatorics of Steenrod operations on the cohomology of Grassmannians
- Lenart
- 1997
(Show Context)
Citation Context ...1 + \Deltas\Deltas\Deltas+ `k \Gammasmk 'wm+k; where negative binomial coefficients are reduced modulo 2. Work on Wu-type formulae appears in [6, 28, 30, 31, 82, 110, 159, 172, 192]. Recently, Lenart =-=[114]-=- has produced integral Wu formulae using differential operators in conjunction with Schur functions, Schubert varieties and the Hammond operators of classical combinatorics. 3.4 Problems Problem 3.6 H... |

4 |
Geometric homology operations
- McCrory
- 1978
(Show Context)
Citation Context ...e shall mention briefly. Early work on the geometric approach to Steenrod squares relates to Smith theory [29, 200, 225], double point sets in bordism with singularities, and SpanierWhitehead duality =-=[130]-=-. Dold sets up the Steenrod algebra for cohomology of a topological space in [60]. Quillen [161] uses Steenrod operations in bordism, further developed by Tom Dieck [202], in connection with formal gr... |

3 |
Operations of the nth kind in K -theory, and what we don't know about
- Adams
- 1974
(Show Context)
Citation Context ...teenrod algebra in section 1 to differential operators. This programme may involve extending the scope of the differential operators Dk to negative k acting on Laurent polynomials in the variables xi =-=[5, 215]-=-. For interpretations of Landweber-Novikov operations as differential operators in the context of conformal field theory, quantum groups and diffeomorphisms of the line, we refer to [33, 34, 35, 98, 1... |

3 |
Monomial bases in the Steenrod algebra
- Arnon
- 1994
(Show Context)
Citation Context ...l showed how these relations lead to a basis of A involving monomials in the Sq2 n , but no closed formulae were proposed for the relations themselves. Since then, several new bases of A have emerged =-=[17, 143, 220]-=-, and closed forms have been discovered for Wall's relations. To describe Wall's basis, let Qnk = Sq2 kSq2k+1 \Deltas\Deltas\DeltasSq2n for n * k. Theorem 1.10 ( Wall [213]) The set of monomials Qn0k0... |

3 |
Une theorie axiomatique des carres de
- Cartan
- 1950
(Show Context)
Citation Context ...erations are called stable. 4s1.1 Early results on the Steenrod algebra Cartan discovered a formula for evaluating a Steenrod square on the cup product of cohomology classes f; g. Theorem 1.1 (Cartan =-=[43]-=-) Sqn(f g) = P0^r^n Sqr(f )Sqn\Gamma r(g). Using earlier work of P.A. Smith and M. Richardson [167] on the homology of cyclic products of topological spaces, Thom [198] and Wu [224, 222] gave a charac... |

3 |
Not the Adem relations
- Crossley, Hubbuck
- 1992
(Show Context)
Citation Context ... conjugation in A have been investigated notably by Don Davis [55] and others [20, 21, 70, 103, 116, 191]. Recent advances in these areas can be traced through the work of Crabb, Crossley and Hubbuck =-=[49, 50, 51, 52, 53]-=-, Ken Monks [139, 138, 142, 143, 140], and Bill Singer and Judith Silverman [174, 175, 178, 182]. In [36], Bullett-Macdonald devise a method for generating the Adem relations by equating coefficients ... |

3 | Some quotients of the Steenrod algebra - Davis - 1981 |

3 |
Triple points of immersed orientable 2n-manifolds in 3n-space
- Eccles, Mitchell
- 1989
(Show Context)
Citation Context ...olynomials divisible by x1 \Deltas\Deltas\Deltasxn. Closed formulae for these actions have been of interest to algebraic and differential topologists, for example in the immersion theory of manifolds =-=[64, 65, 66, 68]-=-. Classically, the Wu formulae [134, 223] answer the question of how the Steenrod squares act on the elementary symmetric polynomials in \Lambda (n) \OmegasF2: Sqk(wm) = wkwm + `k \Gammasm1 'wk\Gamma ... |

3 |
On the weak conjecture on spherical classes
- Hung
- 1996
(Show Context)
Citation Context ...84, 87, 88], but the general problem appears difficult. Hung conjectures that all elements in the positively graded Dickson algebra are hit in W \OmegasF2 if the number of variables is at least three =-=[84, 83, 86, 85]-=-. 61sThe relationship between the Steenrod algebra A and the full matrix semigroups M (n; F2) is similar to the relationship between the differential operator algebra D bears and the symmetric semigro... |

3 |
On a Cartan formula for secondary cohomology operations
- Kristensen
- 1965
(Show Context)
Citation Context ...2 n. Cohen [46] showed how to obtain the Adem relations from the Milnor product formula. Of special interest are the following particular cases. Example 1.11 Sq2m\Gamma 1Sqm = 0 for m ? 0. Kristensen =-=[106, 104, 105]-=- introduced a process which he called differentiation, but which we shall call stripping, for deriving relations from relations in the Steenrod algebra. The stripping process is an action of the dual ... |

3 | Combinatorial Analysis. Reprinted by Chelsea - MacMahon - 1916 |

2 |
A -annihilated elements
- Anick, Peterson
- 1993
(Show Context)
Citation Context ...lgebra acts by Kronecker duality. This action is exploited, for example, in studying the splitting of classifying spaces of general linear groups over finite fields [216], and work on the hit problem =-=[15, 16, 50]-=-. An observation which is sometimes useful in dealing with the action of the Steenrod algebra on W \OmegasF2 is the commutativity with partial differentiation. Lemma 2.21 Sqn @@x i = @ @xi Sq n: This ... |

2 |
Some remarks about symmetric functions
- Brown, Peterson
- 1976
(Show Context)
Citation Context ...n) \OmegasF2: Sqk(wm) = wkwm + `k \Gammasm1 'wk\Gamma 1wm+1 + \Deltas\Deltas\Deltas+ `k \Gammasmk 'wm+k; where negative binomial coefficients are reduced modulo 2. Work on Wu-type formulae appears in =-=[6, 28, 30, 31, 82, 110, 159, 172, 192]-=-. Recently, Lenart [114] has produced integral Wu formulae using differential operators in conjunction with Schur functions, Schubert varieties and the Hammond operators of classical combinatorics. 3.... |

2 |
On the Adem relations
- Cohen
- 1961
(Show Context)
Citation Context ...plied his results to the evaluation of the first few cohomology groups of the Steenrod algebra [213], and to settle some questions raised by Toda [201] about right multiplication in A by Sq2 n. Cohen =-=[46]-=- showed how to obtain the Adem relations from the Milnor product formula. Of special interest are the following particular cases. Example 1.11 Sq2m\Gamma 1Sqm = 0 for m ? 0. Kristensen [106, 104, 105]... |

2 | K-theory and the anti-automorphism of the Steenrod algebra
- Crabb, Crossley, et al.
- 1996
(Show Context)
Citation Context ... conjugation in A have been investigated notably by Don Davis [55] and others [20, 21, 70, 103, 116, 191]. Recent advances in these areas can be traced through the work of Crabb, Crossley and Hubbuck =-=[49, 50, 51, 52, 53]-=-, Ken Monks [139, 138, 142, 143, 140], and Bill Singer and Judith Silverman [174, 175, 178, 182]. In [36], Bullett-Macdonald devise a method for generating the Adem relations by equating coefficients ... |

2 |
Brown{Peterson operations and Steenrod modules’, Quart
- Kane
- 1979
(Show Context)
Citation Context ... The elements D(1p n); D(2pn), as p ranges over primes and n * 0, form a minimal algebraic generating set for D. More details on relations between these generators can be found in [102]. Richard Kane =-=[92]-=- has produced analogues of the Milnor product formulae in BP -theory. Li [115, 116] has constructed global product formulae for certain types of Hopf algebras in terms of the dual coproduct and convol... |

2 |
On the cohomology of two-stage Postnikov systems
- Kristensen
- 1962
(Show Context)
Citation Context ...2 n. Cohen [46] showed how to obtain the Adem relations from the Milnor product formula. Of special interest are the following particular cases. Example 1.11 Sq2m\Gamma 1Sqm = 0 for m ? 0. Kristensen =-=[106, 104, 105]-=- introduced a process which he called differentiation, but which we shall call stripping, for deriving relations from relations in the Steenrod algebra. The stripping process is an action of the dual ... |

1 |
On the structure and applications of the Steenrod algebra
- Abe, algebras
- 1980
(Show Context)
Citation Context |

1 |
On formulae of Thom and
- Adams
- 1961
(Show Context)
Citation Context ...n) \OmegasF2: Sqk(wm) = wkwm + `k \Gammasm1 'wk\Gamma 1wm+1 + \Deltas\Deltas\Deltas+ `k \Gammasmk 'wm+k; where negative binomial coefficients are reduced modulo 2. Work on Wu-type formulae appears in =-=[6, 28, 30, 31, 82, 110, 159, 172, 192]-=-. Recently, Lenart [114] has produced integral Wu formulae using differential operators in conjunction with Schur functions, Schubert varieties and the Hammond operators of classical combinatorics. 3.... |

1 |
The products in the Steenrod rings of the complex and symplectic cobordism theories
- Aikawa
(Show Context)
Citation Context ...phisms. In [102], Kozma extends Landweber's relations, and solves the problem of finding the irreducibles in the Landweber-Novikov algebra over the integers. The same results appear in Aikawa's paper =-=[14]-=-, where an explicit product formula is developed for the additive generators. Recently, a natural product formula has been given in terms of composition of differential operators [221]. 1.3 Remarks A ... |

1 |
An expression for O/(Sqm
- Bausum
- 1975
(Show Context)
Citation Context |

1 |
Stably splitting
- Benson
- 1996
(Show Context)
Citation Context ... of elements in the Steenrod group? Topologists are interested in splitting classifying spaces of finite groups, as we discussed in subsection 7.4. For general information on this subject we refer to =-=[24, 126, 160]-=-. Some recent work on splitting 2-groups can be found in [45]. The following problems include a somewhat self-referential item. 9.3 Problems Problem 9.2 How does the cohomology of the Steenrod group U... |

1 |
Symmetric products and Steenrod squares, Annals of Math. 57
- Bott
- 1953
(Show Context)
Citation Context ...re are a number of other methods for setting up Steenrod operations in various contexts that we shall mention briefly. Early work on the geometric approach to Steenrod squares relates to Smith theory =-=[29, 200, 225]-=-, double point sets in bordism with singularities, and SpanierWhitehead duality [130]. Dold sets up the Steenrod algebra for cohomology of a topological space in [60]. Quillen [161] uses Steenrod oper... |

1 |
Semigroups of maps into groups, operator doubles, and complex bordism
- Buhstaber
- 1995
(Show Context)
Citation Context ...iables xi [5, 215]. For interpretations of Landweber-Novikov operations as differential operators in the context of conformal field theory, quantum groups and diffeomorphisms of the line, we refer to =-=[33, 34, 35, 98, 150]-=-. It would also be interesting to know if the complex bordism of a compact Hausdorff topological space X could be defined in an algebraic manner, perhaps in terms of the algebra of complex-valued func... |

1 |
Some stable splittings of BP for some P of order a power of 2
- Catalano
- 1996
(Show Context)
Citation Context ...ting classifying spaces of finite groups, as we discussed in subsection 7.4. For general information on this subject we refer to [24, 126, 160]. Some recent work on splitting 2-groups can be found in =-=[45]-=-. The following problems include a somewhat self-referential item. 9.3 Problems Problem 9.2 How does the cohomology of the Steenrod group U (1) split as a module over the Steenrod algebra? Problem 9.3... |

1 |
HV is of bounded type over Ap
- Crossley
- 1996
(Show Context)
Citation Context ... conjugation in A have been investigated notably by Don Davis [55] and others [20, 21, 70, 103, 116, 191]. Recent advances in these areas can be traced through the work of Crabb, Crossley and Hubbuck =-=[49, 50, 51, 52, 53]-=-, Ken Monks [139, 138, 142, 143, 140], and Bill Singer and Judith Silverman [174, 175, 178, 182]. In [36], Bullett-Macdonald devise a method for generating the Adem relations by equating coefficients ... |

1 |
On the height of Sq2 n
- Davis
- 1985
(Show Context)
Citation Context ...height is known [116, 139, 209]. For example, the Wilson conjecture has been proved. 49sTheorem 6.3 (Walker-Wood [209]) The nilpotence height of Sq2 k is 2k + 2. Previously, it was shown by Don Davis =-=[55, 57]-=- that (Sq2 k )2k+1 6= 0. This can be seen directly by evaluation on a monomial in two variables in the polynomial algebra F2[x; y]. Theorem 6.4 (Walker and Wood [212]) (Sq2 n)2n+1(xyk2n+1\Gamma 1) = h... |

1 | Lots of Hopf algebras
- Duflot
- 1996
(Show Context)
Citation Context ...on that if an element E in A has excess k, then E(x1 \Deltas\Deltas\Deltasxm) 6= 0 for all m * k. 2.4 Other sources of integral Steenrod squares In the theory of deformations of Hopf algebras, Duflot =-=[63]-=- introduces an integral version of the Steenrod algebra, starting with the free associative algebra V = Z[X1; X2; : : : Xn; : : :] on a countable number of generators Xi with X0 = 1. Then V is a Hopf ... |

1 |
Splitting \Sigma (CP 1 \Theta CP 1) localised at 2, Quart
- Eccles, Mitchell
- 1988
(Show Context)
Citation Context ...f *. There are analogues for splitting the stable type of p-localisations of the product of n copies of infinite complex projective space, otherwise known as the classifying space BT n of the n-torus =-=[39, 67, 79, 135, 136, 216, 217]-=-. Little is known about the odd prime analogue of the first occurrence problem, although a few cases are resolved [40, 41, 62, 216]. The first occurrence of a simple module as a submodule is also an i... |

1 |
H\Lambda (RP 1 \Theta \Delta \Delta \Delta \Theta RP 1) as a module over the Steenrod algebra
- Giambalvo, Hung, et al.
- 1994
(Show Context)
Citation Context |

1 |
On the height of Sq2 n, Contemp
- Giambalvo, Peterson
- 1995
(Show Context)
Citation Context ...(Sq2 k)2k+1 cannot be detected on polynomials in one variable because, for any positively graded element E in the Steenrod algebra, E2 annihilates all polynomials in a single variable. An argument in =-=[73]-=-, prior to the general proof, had shown directly that (Sq2 k)2k+2 vanishes on polynomials in two variables. This raises the general question of how few variables are needed to detect a non-zero elemen... |

1 |
On Wu’s formula of Steenrod squares on Stiefel{Whitney classes
- Hsiang
- 1963
(Show Context)
Citation Context |

1 |
The nilpotent height of P st for odd primes
- Karaca
- 1996
(Show Context)
Citation Context ...212]) The nilpotence height of P p n is p(n+1). Monks has generalised Theorem 6.4 to the P st family for the prime 2 in [139], and Karaca has extended this to the odd prime case. Theorem 6.6 ( Karaca =-=[94]-=-) In the modulo p Steenrod algebra the nilpotence height of P st is p[s=t] + p: The work of Monks contains further information on nilpotence heights of squares with odd exponent and Milnor basis eleme... |

1 |
Formes diff'erentiales non-commutatives et op'erations de Steenrod, Topology 34
- Karoubi
- 1995
(Show Context)
Citation Context ..., further developed by Tom Dieck [202], in connection with formal groups and the Lazard ring. In [25], Bisson and Joyal treat similar topics in terms of a divided differential theory of coverings. In =-=[95, 96]-=-, Karoubi combines the de Rham theory of differential forms for the homology of manifolds with symmetric products of topological spaces, to obtain certain chain complexes in terms of which cohomology ... |

1 |
Formes diff'erentiales non-commutatives et cohomologie `a coefficients arbitraires
- Karoubi
- 1995
(Show Context)
Citation Context ..., further developed by Tom Dieck [202], in connection with formal groups and the Lazard ring. In [25], Bisson and Joyal treat similar topics in terms of a divided differential theory of coverings. In =-=[95, 96]-=-, Karoubi combines the de Rham theory of differential forms for the homology of manifolds with symmetric products of topological spaces, to obtain certain chain complexes in terms of which cohomology ... |

1 |
The Steenrod algebra action on generators of subgroups of GLn(Z=pZ
- Kechagias
- 1993
(Show Context)
Citation Context ...e Dickson invariants. The action of the Steenrod squares on the Dickson invariants and, more generally, invariants of parabolic subgroups of general linear groups, is worked out in a number of places =-=[9, 81, 83, 99, 146, 193, 214]-=-, and global product formulae are produced in [116]. The restricted hit problem for the Dickson algebra is solved for a small number of variables in [72, 84, 87, 88], but the general problem appears d... |

1 |
On the coecients of the double total squaring operation
- Klippenstein, Lomonaco
- 1992
(Show Context)
Citation Context ..., 116] has constructed global product formulae for certain types of Hopf algebras in terms of the dual coproduct and convolution of formal sequences. Total Steenrod squaring operations are studied in =-=[100, 120]-=-. Aikawa's product formula [14] for the Landweber-Novikov algebra is phrased in the traditional sequence notation s(r1; r2; : : :). In [221], the product formula 17sin the differential operator algebr... |

1 |
Irreducibles in the Landweber{Novikov algebra
- Kozma
- 1976
(Show Context)
Citation Context ...v expresses the Steenrod operations and MU operations by their action on polynomials, and ties in the theory with symmetric functions, Stiefel-Whitney classes, Chern classes and Thom isomorphisms. In =-=[102]-=-, Kozma extends Landweber's relations, and solves the problem of finding the irreducibles in the Landweber-Novikov algebra over the integers. The same results appear in Aikawa's paper [14], where an e... |

1 |
Formulas for Brown{Peterson operations
- Li
- 1994
(Show Context)
Citation Context ...mal algebraic generating set for D. More details on relations between these generators can be found in [102]. Richard Kane [92] has produced analogues of the Milnor product formulae in BP -theory. Li =-=[115, 116]-=- has constructed global product formulae for certain types of Hopf algebras in terms of the dual coproduct and convolution of formal sequences. Total Steenrod squaring operations are studied in [100, ... |

1 |
Generalisation of Peterson’s and other formulas’, Adv
- Lin
- 1985
(Show Context)
Citation Context ...or basis elements Sq(0; r2; r3; : : :). As a simple illustration of the stripping technique, we re-work an old result of Mizuno and Saito [137], extended by Peterson [157] and further extended by Lin =-=[117]-=- and Li [116], which states the following relations in the Steenrod algebra. Example 5.7 X 0^i^k SqiSqk\Gamma i = 0; k 6= 0 mod 3;X 0^i^k SqiSqk\Gamma i = Sq(0; k=3); k = 0 mod 3: Consider the vector ... |

1 |
The complete Steenrod algebra and the generalised Dickson algebra’, Algebraic topology: new trends in localization and periodicity (San Feliu de Guixols
- Llerena, Hung
- 1994
(Show Context)
Citation Context ... fields, Mui [146] identifies the Milnor basis elements in terms of Dickson invariants under a certain coaction of the cohomology of elementary 2-groups. Recent work in this direction can be found in =-=[118]-=-. Lomonaco [119] has further related the Dickson invariants to the Dyer-Lashof algebra in terms of May's universal Steenrod algebra [128] and the lambda algebra, which is a powerful device for calcula... |

1 |
A basis of admissible monomials for the universal Steenrod algebra’, Ricerche Mat
- Lomonaco
- 1991
(Show Context)
Citation Context ...6] identifies the Milnor basis elements in terms of Dickson invariants under a certain coaction of the cohomology of elementary 2-groups. Recent work in this direction can be found in [118]. Lomonaco =-=[119]-=- has further related the Dickson invariants to the Dyer-Lashof algebra in terms of May's universal Steenrod algebra [128] and the lambda algebra, which is a powerful device for calculating the cohomol... |

1 |
The iterated total squaring operation
- Lomonaco
- 1992
(Show Context)
Citation Context ..., 116] has constructed global product formulae for certain types of Hopf algebras in terms of the dual coproduct and convolution of formal sequences. Total Steenrod squaring operations are studied in =-=[100, 120]-=-. Aikawa's product formula [14] for the Landweber-Novikov algebra is phrased in the traditional sequence notation s(r1; r2; : : :). In [221], the product formula 17sin the differential operator algebr... |

1 |
Classifying spaces and their maps’, Homotopy theory and its applications (Cocoyoc
- Martino
- 1993
(Show Context)
Citation Context ... of elements in the Steenrod group? Topologists are interested in splitting classifying spaces of finite groups, as we discussed in subsection 7.4. For general information on this subject we refer to =-=[24, 126, 160]-=-. Some recent work on splitting 2-groups can be found in [45]. The following problems include a somewhat self-referential item. 9.3 Problems Problem 9.2 How does the cohomology of the Steenrod group U... |

1 |
A general algebraic approach to Steenrod operations, The Steenrod Algebra and its
- Matsumura, algebra, et al.
- 1980
(Show Context)
Citation Context |