Abstract
The mod 2 Steenrod algebra \(\mathcal {A}_2\) can be defined as the quotient of the mod 2 Leibniz–Hopf algebra \(\mathcal {F}_2\) by the Adem relations. Dually, the mod 2 dual Steenrod algebra \(\mathcal {A}_2^*\) can be thought of as a sub-Hopf algebra of the mod 2 dual Leibniz–Hopf algebra \(\mathcal {F}_2^*\). We study \(\mathcal {A}_2^*\) and \(\mathcal {F}_2^*\) from this viewpoint and give generalisations of some classical results in the literature.
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1 The mod 2 Leibniz–Hopf algebra and its dual
Let \(\mathcal {F}_2\) be the free associative algebra over \(\mathbb {F}_2\) generated by the indeterminates \(S^1, S^2, S^3, \ldots \) of degree \(|S^i|=i\). We often denote the unit 1 by \(S^0\). This algebra is equipped with a co-commutative co-product given by
which makes it a graded connected Hopf algebra. This algebra \(\mathcal {F}_2\) is often called the mod 2 Leibniz–Hopf algebra. As an \(\mathbb {F}_2\)-module, \(\mathcal {F}_2\) has the following canonical basis:
where we regard \(S^I=1\) when \(n=0\).
Note that the integral counterpart of \(\mathcal {F}_2\) is called the Leibniz–Hopf algebra and is isomorphic to the ring of non-commutative symmetric functions [7] and the Solomon Descent algebra [17]. Its graded dual is the ring of quasi-symmetric functions with the outer co-product, which has been studied by Hazewinkel, Malvenuto, and Reutenauer in [8,9,10,11,12].
The mod 2 Steenrod algebra \(\mathcal {A}_2\) is defined to be the quotient Hopf algebra of \(\mathcal {F}_2\) by the ideal generated by the Adem relations:
Denote the quotient map by \(\pi : \mathcal {F}_2\rightarrow \mathcal {A}_2\) and \(Sq^i=\pi (S^i)\). It is well-known (see, for example, [18]) that the admissible monomials
form a module basis for \(\mathcal {A}_2\). We will adhere to this purely algebraic definition and will not use any other known facts about \(\mathcal {A}_2\).
By taking the graded dual of \(\pi \), we obtain the following inclusion of Hopf algebras
\(\mathcal {F}_2^*\) is given a module basis \(S_I\) dual to \(S^I\), that is,
Similarly, we have the dual basis \(\{Sq_J \mid J \text { admissible} \}\) for \(\mathcal {A}_2^*\) determined by
The commutative product among the basis elements in \(\mathcal {F}_2^*\) is given by the overlapping shuffle product (see §2) and the co-product is given by
The purpose of this paper is to deduce some of the classical results on \(\mathcal {A}_2^*\) and its generalisations by considering it as a subalgebra of \(\mathcal {F}_2^*\). We are particularly interested in the following problems.
Problem 1
-
(i)
Determine the coefficients in
$$\begin{aligned} \pi ^*(Sq_J) = \sum _{I} C^I_J S_I \end{aligned}$$(4)for all admissible sequences J. This is important since in the dual it is equivalent to computing the coefficients of the Adem relations
$$\begin{aligned} Sq^I = \sum _{J:\text {admissible}} C^I_J Sq^J \end{aligned}$$(5)for all sequences I.
-
(ii)
Give an expansion of the dual Milnor bases in terms of the dual admissible monomial bases, i.e., determine the coefficient \(B^L_J\) in
$$\begin{aligned} \xi ^L = \sum _{J:\text {admissible}} B^L_J Sq_J, \end{aligned}$$where \(\xi _n=Sq_{2^{n-1},2^{n-2}\cdots 2^1,2^0}\) and \(\xi ^L=\xi _1^{l_1}\xi _2^{l_2}\cdots \xi _n^{l_n}\) for \(L=(l_1,l_2,\ldots , l_n)\).
-
(iii)
Generalise Milnor’s conjugation formula [13] in \(\mathcal {A}_2^*\) to \(\mathcal {F}_2^*\). The formula for \(\mathcal {A}_2^*\) is:
$$\begin{aligned} \chi (\xi _n)= \sum _\alpha \prod _{i=1}^{l(\alpha )} \xi _{\alpha (i)}^{2^{\sigma (i)}}, \end{aligned}$$where \(\alpha =(\alpha (1)|\alpha (2)|\ldots |\alpha (l(\alpha ))\) runs through all the compositions of the integer n and \(\sigma (i)=\sum _{j=1}^{i-1} \alpha (j)\).
Several different methods are known for resolving (i) and (ii) (see for example, [15, 19]), but our argument (Sect. 4) is new in that it is purely combinatorial using the overlapping shuffle product on \(\mathcal {F}_2^*\). We implemented our algorithm into a Maple code [16]. In Sect. 5 we discuss the conjugation (or antipode) in \(\mathcal {F}_2^*\) and give an answer to (iii). Finally, we give an explicit duality between the conjugation invariants in \(\mathcal {F}_2\) and \(\mathcal {F}_2^*\) in Sect. 6.
2 Overlapping Shuffle product
We recall the definition of the overlapping shuffle product ([2, Section 2],[8]). Let \(\mathcal {W}\) be the set of finite sequences of natural numbers:
Note that we allow the length 0 sequence. Consider the \(\mathbb {F}_2\)-module \(\mathbb {F}_2\langle \mathcal {W}\rangle \) freely generated by \(\mathcal {W}\). For a sequence \(I=(i_1,i_2,\ldots ,i_n)\), denote its tail partial sequence \((i_k,i_{k+1},\ldots ,i_n)\) by \(I_k\). When \(n<k\), we regard \(I_k\) as the length 0 sequence. We use the convention
The overlapping shuffle product on \(\mathbb {F}_2\langle \mathcal {W}\rangle \) is defined as follows:
Definition 1
For \(A=(a_1,a_2,\ldots ,a_n)\) and \(B=(b_1,b_2,\ldots ,b_m)\), define their product inductively by
The product on \(\mathbb {F}_2\langle \mathcal {W}\rangle \) is defined by the linear extension of the above.
We say a term in \(A\cdot B\) is a-first if there exists k such that \(a_k\) goesFootnote 1 to an entry to the left of \(b_k\) and \(a_i\) goes to the same entry as \(b_i\) (that is, the entry makes \(a_i+b_i\)) for all \(i<k\). For example, \((a_1+b_1,a_2,b_2,b_3,a_3)\) is a-first while \((a_1+b_1,b_2,a_2,a_3,b_3)\) is not. Observe that
Lemma 1
For equal length sequences, we have
where Z is a sum of a-first terms and \(\tau \) flips the occurrence of \(a_i\) and \(b_i\) for all i. In particular, the product is commutative.
Example 1
where the second line consists of a-first terms and the third line is the \(\tau \)-image of the second line.
Corollary 1
For \(A=(a_1,\ldots ,a_n)\),
Proof
In this case, the flip map \(\tau \) in Lemma 1 is the identity. \(\square \)
It is easy to see from the duality relation \(\langle S_I S_J, S^K \rangle =\langle S_I\otimes S_J, \Delta (S^K) \rangle \) that the product on \(\mathcal {F}_2^*\) dual to (1) is given by \(S_I S_J= \sum _{K\in I\cdot J} S_K\).
3 Dual Steenrod algebra as a sub-Hopf algebra of \(\mathcal {F}_2^*\)
To identify the image of the inclusion \(\pi ^*:\mathcal {A}_2^* \rightarrow \mathcal {F}_2^*\), we prove some lemmas in this section. Let \(\xi _n=Sq_{2^{n-1}, 2^{n-2}, \ldots , 2^0}\).
Lemma 2
Proof
For the first equation, we have to show that for any non-admissible sequence I, the right-hand side of
does not contain \(Sq^{2^n}\). If there exists such an I, we can assume it has length two, that is, \(I=(i,j)\). (Because the right-hand side is obtained by successively applying the length two relations.) By the Adem relations in Eq. (2), we have \(i+j=2^n\) and
However, the binary expressions of \(2^n-1-i\) and i are complementary and the binary expression of \(2^n-1-i\) contains at least one digit with 0. Hence, by Lucas’ Theorem, we have \(\left( {\begin{array}{c}2^n-1-i\\ i\end{array}}\right) \equiv 0 \mod 2\); we arrive at a contradiction.
For the second equation, suppose that there exists an \(I=(i,j)\) such that \(i < 2j\) and
contains \(Sq^{2^{n-k}}\) or \(Sq^{2^{n-k}}Sq^{2^{n-k-1}}\) as a summand. The former case is already ruled out by the first equation. For the latter case to happen, we should have
But this implies \(j\le 2^{n-k-1}\) so \(i\ge 2j\); we arrive at a contradiction. \(\square \)
Put \(\bar{\xi }_n=\pi ^*(\xi _n) = S_{2^{n-1}, 2^{n-2}, \ldots , 2^0}\). We denote by \(\widetilde{\mathcal {A}}_2^*\) the subalgebra of \(\mathcal {F}_2^*\) generated by \(\{\bar{\xi }_n \mid 0<n \}\). For a sequence \(L=(l_1,l_2,\ldots ,l_n)\) of non-negative integers, we denote \(\bar{\xi }_1^{l_1} \bar{\xi }_2^{l_2} \cdots \bar{\xi }_n^{l_n}\) by \(\bar{\xi }^L\). Then, the monomials \(\bar{\xi }^L\) span \(\widetilde{\mathcal {A}}_2^*\). Now, we identify \(\widetilde{\mathcal {A}}_2^*\) with \(\mathrm {Im}(\pi ^*)\).
Recall the definition of the excess vector of an admissible sequence \(J=(j_1,j_2,\ldots ,j_n)\):
This gives a bijection between admissible sequences and sequences of non-negative integers. The inverse is given by
We put the right lexicographic order on \(\mathcal {W}\), i.e.,
This induces an ordering on the basis elements \(S_I\) which is compatible with the overlapping shuffle product. Observe that the lowest term in the product \(S_I \cdot S_{I'}\) for \(I=(i_1,i_2,\ldots )\) and \(I'=(i'_1,i'_2,\ldots )\) is \(S_{(i_1+i'_1,i_2+i'_2,\ldots )}\).
Lemma 3
For an admissible sequence J,
Proof
We proceed by induction on \(J=(j_1,\ldots ,j_n)\). Put \(J'=(j_1-2^{n-1},j_2-2^{n-2},\ldots ,j_n-2^0)\). Then by induction hypothesis,
It follows that
\(\square \)
By this upper-triangularity, the monomials \(\bar{\xi }^L\) are linearly independent and we have
Theorem 1
Proof
By Lemma 3 in each degree \(\widetilde{\mathcal {A}}_2^*\) has the same dimension as \(\mathcal {A}_2^*\) (the number of admissible sequences). \(\square \)
This is nothing but the well-known fact:
Corollary 2
[13]
where
4 Computation with \(\pi ^*\)
Recall from [19, Section 4] the linear left inverse \(r:\mathcal {F}_2^* \rightarrow \mathcal {A}_2^*\) of \(\pi ^*\):
For (ii) of Problem 1, we can compute
and it reduces to computing admissible sequences occurring in the overlapping shuffle product.
For (i) of Problem 1, by Corollary 2 we have
and the left-hand side can be computed by the overlapping shuffle product. Thus, we can compute inductively the coefficients \(C^I_J\) in
We implemented the algorithm into a Maple code [16].
Example 2
We demonstrate the above algorithm in low degrees. First, compute \(\pi ^*\)-image of monomials \(\xi ^L\):
Taking r on the both sides of equations, we obtain
Again taking \(\pi ^*\) on the both sides of the equations, we obtain
Finally, by using the upper-triangularity, we obtain
5 Formula for the conjugation
Any connected commutative or co-commutative Hopf algebra has a unique conjugation \(\chi \) satisfying
where \(\Delta (x)=\sum x'\otimes x''\) and \(\deg (x)>0\) [14]. The conjugation invariants in \(\mathcal {A}_2^*\) is studied in [5] because it is relevant to the commutativity of ring spectra [1, Lecture 3]. The same problem in \(\mathcal {F}_2^*\) has been also studied in [3, 4]. Here we investigate them through our point of view.
Since \(\pi ^*\) is a Hopf algebra homomorphism, we have \(\pi ^*\circ \chi _{\mathcal {A}_2^*} = \chi _{\mathcal {F}_2^*}\circ \pi ^*\), where \(\chi _{\mathcal {A}_2^*}\) and \(\chi _{\mathcal {F}_2^*}\) denote the conjugation operations in \(\mathcal {A}_2^*\) and \(\mathcal {F}_2^*\) respectively. For the module basis \(S_I\) in \(\mathcal {F}_2^*\), the conjugation \(\chi _{\mathcal {F}_2^*}\) is calculated combinatorially.
Definition 2
The coarsening set C(I) of a sequence \(I=(i_1,\ldots ,i_l)\) is defined recursively as
where \(I'_2\) is the tail partial sequence \((i'_2,\ldots ,i'_{l'})\) of \(I'=(i'_1,i'_2,\ldots ,i'_{l'})\).
Example 3
\(C((a,b,c))=\{ (a,b,c), (a+b,c), (a,b+c), (a+b+c) \}\).
A formula for the conjugation operation in the dual Leibniz–Hopf algebra is given by Ehrenborg [6, Proposition 3.4]. We now give a simple proof for its mod 2 reduction.
Proposition 1
where \(I^{-1}=(i_l,\ldots ,i_1)\) is the reverse sequence of \(I=(i_1,\ldots ,i_l)\).
Proof
The conjugation is uniquely characterised by
where \(\Delta (x)=\sum x'\otimes x''\) and \(\deg (x)>0\). We put \(\chi '(S_I)=\sum _{I'\in C(I^{-1})} S_{I'}\) and show that it satisfies the above equations. It is obvious that \(\chi '(1)=1\). Since the co-product is given in (3), the second equation reads
We regard an element of \(\mathbb {F}_2\langle \mathcal {W}\rangle \) with a finite subset of \(\mathcal {W}\) in the obvious way. We investigate relation between coarsening and the overlapping shuffle product. Define
We observeFootnote 2 that \(C(I^{-1}) \subset C_1(I)\) and \(C'_1(I):=C_1(I) {\setminus } C(I^{-1})\) consists of those sequences that \(i_1\) appears to the left of \(i_2\). In turn, \(C'_1(I) \subset C_2(I)\) and \(C'_2(I):=C_2(I) {\setminus } C'_1(I)\) consists of those sequences that \(i_2\) appears to the left of \(i_3\). Continuing similarly, we obtain
It follows that
and \(\sum _{k=0}^{l} S_{i_1,\ldots ,i_k} \chi '(S_{i_{k+1},\ldots ,i_n})=0\). \(\square \)
We give another formula for \(\chi _{\mathcal {F}_2^*}(S_I)\).
Definition 3
For a sequence \(a_1,a_2,\ldots ,a_n\), the set of ordered block partitions \(\mathcal {P}(a_1,a_2,\ldots ,a_n)\) consists of elements of the form
where \(1\le i_1<i_2<\cdots <i_l=n\). We denote \(l(\beta )=l\) and \(\beta (k)=(a_{i_{k-1}+1},\ldots ,a_{i_k})\). Or inductively, we can define
Theorem 2
Proof
Let \(I=(a_1,a_2,\ldots ,a_n)\). Put
and we check that
Then, by the uniqueness of the conjugation, we have \(\chi _{\mathcal {F}_2^*}=\chi '\). The first assertion is trivial. For the second, observe that by (7)
Hence, we have
\(\square \)
Example 4
The first line is computed by Proposition 1, and the second by Theorem 2.
Theorem 2 can be thought of as a generalisation of Milnor’s conjugation formula in \(\mathcal {A}_2^*\). To see this, we first show a small lemma:
Lemma 4
Proof
This is a direct consequence of Corollary 1 combined with Lemma 2. \(\square \)
Corollary 3
[13, Lemma 10]
where \(\alpha =(\alpha (1)|\alpha (2)|\ldots |\alpha (l(\alpha ))\) runs through all the compositions of the integer n and \(\sigma (k)=\sum _{j=1}^{k-1} \alpha (j)\).
Proof
We apply the injection \(\pi ^*\) to the both sides of (8) and show that they coincide. For the left-hand side, by Lemma 4 we have
Since \(\pi ^*(\xi _{\alpha (k)}^{2^{\sigma (k)}})=S_{2^{\alpha (k)+\sigma (k)-1},\ldots ,2^{\sigma (k)}}\) by Lemma 4, we see
So when \(\alpha \) ranges over all compositions of n, we get all the ordered block partitions of the sequence \(2^{n-1},2^{n-2},\ldots ,2^0\). The assertion follows from Theorem 2. \(\square \)
6 Duality between \(\mathcal {F}_2\) and \(\mathcal {F}_2^*\)
In the previous section we discussed how to compute the conjugation in \(\mathcal {F}_2^*\). Here, we relate the conjugation in \(\mathcal {F}_2\) with that in \(\mathcal {F}_2^*\) by using a self-duality of \(\mathcal {W}\). Denote \(I \preceq I'\) if \(I\in C(I')\). We think of \(I\in \mathcal {W}\) as a string of 1’s separated by ‘\(+\)’ and commas; \((\underbrace{1+1+\cdots +1}_{i_1}, \underbrace{1+1+\cdots +1}_{i_2},\ldots ,\underbrace{1+1+\cdots +1}_{i_l})\).
Definition 4
We define the dual \(\bar{I}\in \mathcal {W}\) of I by switching \(+\) and the commas.
Example 5
For \(I=(1,3,2)=(1,1+1+1,1+1)\), its dual is
It is easily seen that \(\bar{\bar{I}}=I\) and \(I \preceq I' \Leftrightarrow \bar{I} \succeq \bar{I}'\). Extend the duality to one between \(\mathcal {F}_2\) and \(\mathcal {F}_2^*\) by
Theorem 3
We have \(D\circ \chi _{\mathcal {F}_2}= \chi _{\mathcal {F}_2^*}\circ D\). In particular, \(f \in \mathcal {F}_2\) is a conjugation invariant if and only if so is \(\bar{f} \in \mathcal {F}_2^*\).
Proof
We compute
Put \(\chi '(S^I)=\sum _{I'\succeq I^{-1}} S^{I'}\). Then, one can check \(\chi '(1)=1\) and \(\sum x' \chi '(x'')=0\) for \(\Delta x= \sum x'\otimes x''\) as in Proposition 1. Hence, by the uniqueness of the conjugation, we have \(\chi '=\chi _{\mathcal {F}_2}\). \(\square \)
Example 6
\(f=S^{1,1,2}+S^{2,1,1}+S^{1,1,1,1}\) is a \(\chi _{\mathcal {F}_2}\)-invariant, whilst \(D(f)=S_{3,1}+S_{1,3}+S_4\) is a \(\chi _{\mathcal {F}_2^*}\)-invariant.
Remark 1
The sub-module of the conjugation invariants in \(\mathcal {F}_2\) is \(\ker (\chi _{\mathcal {F}_2}-1)\) and that in \(\mathcal {F}_2^*\) is \(\ker (\chi _{\mathcal {F}_2^*}-1)\). The conjugations in \(\mathcal {F}_2\) and \(\mathcal {F}_2^*\) are dual to each other, and hence, the linear map \(\chi _{\mathcal {F}_2}-1\) is transpose to \(\chi _{\mathcal {F}_2^*}-1\) with the kernel of same dimension [3]. Theorem 3 gives more information by specifying an explicit correspondence between their elements.
Notes
When calculated symbolically.
Here, we deal with sequences symbolically so that we avoid cancellations like \((i_3+i_2,i_1)+(i_3+i_1,i_2)=0\) when \(i_1=i_2\).
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Acknowledgements
We would like to thank Stephen Theriault, Martin Crossley, and Carmen Rovi for their comments on the earlier version of this paper.
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Communicated by Stewart Priddy.
The second named author was partially supported by KAKENHI, Grant-in-Aid for Young Scientists (B) 26800043 and JSPS Postdoctoral Fellowships for Research Abroad.
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Turgay, N.D., Kaji, S. The mod 2 dual Steenrod algebra as a subalgebra of the mod 2 dual Leibniz-Hopf algebra. J. Homotopy Relat. Struct. 12, 727–739 (2017). https://doi.org/10.1007/s40062-016-0163-x
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DOI: https://doi.org/10.1007/s40062-016-0163-x