Admissible Monomials Research Articles

Products of Admissible Monomials in the Polynomial Algebra as a Module over the Steenrod Algebra

Let ${\P}(n) ={\F}[x_1,\ldots,x_n]$ be the polynomial algebra in $n$ variables $x_i$, of degree one, over the field $\F$ of two elements. The mod-2 Steenrod algebra $\A$ acts on ${\P }(n)$ according to well known rules. A major problem in algebraic topology is that of determining $\A^+{\P}(n)$, the image of the action of the positively graded part of $\A$. We are interested in the related problem of determining a basis for the quotient vector space ${\Q}(n) = {\P}(n)/\A^{+}\P(n)$. Both ${\P }(n) =\bigoplus_{d \geq 0} {\P}^{d}(n)$ and ${\Q}(n)$ are graded, where ${\P}^{d}(n)$ denotes the set of homogeneous polynomials of degree $d$. ${\Q}(n)$ has been explicitly calculated for $n=1,2,3,4$ but problems remain for $n \geq 5.$ In this note we show that if $u = x_{1}^{m_1} \cdots x_{k}^{m_{k}} \in {\P}^{d}(k)$ and $v = x_{1}^{e_1} \cdots x_{r}^{e_{r}} \in {\P}^{d'}(r)$ are an admissible monomials, (that is, $u$ and $v$ meet a criterion to be in a certain basis for ${\Q}(k)$ and ${\Q}(r)$ respectively), then for each permutation $\sigma \in S_{k+r}$ for which $\sigma(i)<\sigma(j),$ $i<j\leq k$ and $\sigma(s)<\sigma(t),$ $k<s<t\leq k+r,$ the monomial $x_{\sigma(1)}^{m_1} \cdots x_{\sigma(k)}^{m_{k}} x_{\sigma(k+1)}^{e_1} \cdots x_{\sigma(k+r)}^{e_r} \in {\P}^{d+d'}(k+r)$ is admissible. As an application we consider a few cases when $n=5.$

A REMARK ON THE CONJUGATION IN THE STEENROD ALGEBRA

Abstract. We investigate the Hopf algebra conjugation, χ, of the mod2 Steenrod algebra, A 2 , in terms of the Hopf algebra conjugation, χ ′ , ofthe mod 2 Leibniz–Hopf algebra. We also investigate the fixed points ofA 2 under χ and their relationship to the invariants under χ ′ . 1. IntroductionThe mod 2 Steenrod algebra A 2 is the free associative graded algebra gen-erated by the Steenrodsquares Sq n [18] of degree n, n ≥ 1, over F 2 subject tothe AdemrelationsSq a Sq b = ⌊ X a/2⌋s=0 b−1−sa−2sSq a+b−s Sq s for 0 < a < 2b.Conventionally, Sq 0 = 1, the multiplicative identity. Topologically, A 2 is thealgebra of stable cohomology operations for ordinary cohomology H ∗ over F 2 .A monomial in A 2 can be written in the form Sq j 1 Sq j 2 ···Sq j r , which weshall denote by Sq j 1 ,j 2 ,...,j r . Admissible monomials form a vector space ba-sis “admissible basis” for A 2 . Milnor [16] determined the graded connectedHopf algebra structure of A 2 by a cocomutative coproduct given by ∆(Sq

Some Relations between Admissible Monomials for the Polynomial Algebra

LetP(n)=F2[x1,…,xn]be the polynomial algebra innvariablesxi, of degree one, over the fieldF2of two elements. The mod-2 Steenrod algebraAacts onP(n)according to well known rules. A major problem in algebraic topology is of determiningA+P(n), the image of the action of the positively graded part ofA. We are interested in the related problem of determining a basis for the quotient vector spaceQ(n)=P(n)/A+P(n).Q(n)has been explicitly calculated forn=1,2,3,4but problems remain forn≥5. BothP(n)=⨁d≥0Pd(n)andQ(n)are graded, wherePd(n)denotes the set of homogeneous polynomials of degreed. In this paper, we show that ifu=x1m1⋯xn-1mn-1∈Pd′(n-1)is an admissible monomial (i.e.,umeets a criterion to be in a certain basis forQ(n-1)), then, for any pair of integers (j,λ),1≤j≤n, andλ≥0, the monomialhjλu=x1m1⋯xj-1mj-1xj2λ-1xj+1mj⋯xnmn-1∈Pd′+(2λ-1)(n)is admissible. As an application we consider a few cases whenn=5.

Linking first occurrence polynomials over 𝔽pby Steenrod operations

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.

Linking First Occurrence Polynomials over F2 by Steenrod Operations

It is proved that for every irreducible representation L (λ) of the full matrix semigroup M n ( F 2 ), the first occurrence of L (λ) as a composition factor in the polynomial algebra P = F 2 [ x 1 , …, x n ] is linked by a Steenrod operation to the first occurrence of L (λ) as a submodule in P . This Steenrod operation is given explicitly as the image of an admissible monomial in the Steenrod squares Sq r under the canonical anti-automorphism χ of the mod 2 Steenrod algebra A . The first occurrences of both kinds are also linked to higher degree occurrences of L (λ) by elements of the Milnor basis of A .

Formula omitted]-subalgebras, Milnor basis, and cohomology of Eilenberg-MacLane spaces

We describe mod p cohomology rings of Eilenberg-MacLane spaces in terms of the Milnor basis rather than in terms of admissible monomials of the Steenrod algebra. We give a formula of excess for the Milnor basis elements, correcting Kraines' formula in odd prime case. Using the Milnor basis description, we study and characterize certain polynomial subalgebras generated by elements obtained by applying maximum number of Milnor primitives on mod p fundamental classes of Eilenberg-MacLane spaces. A simple and interesting unstable pattern emerges. These subalgebras are exact images of the BP-Thom map into mod p cohomology rings.

Split dual Dyer-Lashof operations

For each admissible monomial of Dyer-Lashof operations Q I , we define a corresponding natural function Q ̂ I:T H ̄ ∗(X)→H ∗(Ω nΣ nX) , called a Dyer-Lashof splitting. For every homogeneous class x in H ∗(X) , a Dyer-Lashof splitting Q ̂ I determines a canonical element y in H ∗(Ω nϵ nX) so that y is connected to x by the dual homomorphism to the operation Q I . The sum of the images of all the admissible Dyer-Lashof splittings contains a complete set of algebra generators for H ∗(Ω nΣ nX) .

Change of basis, monomial relations, and Pts bases for the Steenrod algebra

The relationship between several common bases for the mod 2 Steenrod algebra is explored and a family of bases consisting of monomials in distinct P t s 's is developed. A recursive change of basis formula is produced to convert between the Milnor basis and each of the bases for which the change of basis matrix in every grading is upper triangular. In particular, it is shown that the basis of admissible monomials, the P t s bases, and two bases due to Arnon, are all bases having this property, and the corresponding change of basis formula is produced for each of them. Some monomial relations for the mod 2 Steenrod algebra are then obtained by exploring the change of basis transformations.

Stability of the null solution of parabolic functional inequalities

Uniqueness and stability theorems are established for coupled systems of parabolic differential equations which may involve a Volterra-type dependence on the past history of the process. We allow retarded or deviating arguments, convolution-type memory terms, and strong coupling. (This means that all the space derivatives up to a given order can occur in all the equations.) Our results for strong coupling depend on the concept of “admissible monomial” which is here introduced for the first time and has no counterpart in the linear case. It is possible for uniqueness to fail in general, but to be restored (relative to a tolerably large class of functions of ( x , t ) (x,t) ) if a single solution independent of x exists. Another curious feature of these theorems, depending again on the concept of admissible monomial, is that conditions for uniqueness can involve derivatives of order much higher than those occurring in the equation. Examples given elsewhere show that the results are, in various respects, sharp. Thus, the seemingly peculiar hypotheses do not arise from deficient technique, but from the actual behavior of strongly coupled systems. The paper concludes with a new method of dealing with unbounded regions for the difficult case in which the functional occurs in the boundary operator as well as in the differential equation.