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Tag: Manin

The F_un folklore

All esoteric subjects have their own secret (sacred) texts. If you opened the Da Vinci Code (or even better, the original The Holy blood and the Holy grail) you will known about a mysterious collection of documents, known as the “Dossiers secrets“, deposited in the Bibliothèque nationale de France on 27 April 1967, which is rumoured to contain the mysteries of the Priory of Sion, a secret society founded in the middle ages and still active today…

The followers of F-un, for $\mathbb{F}_1 $ the field of one element, have their own collection of semi-secret texts, surrounded by whispers, of which they try to decode every single line in search of enlightenment. Fortunately, you do not have to search the shelves of the Bibliotheque National in Paris, but the depths of the internet to find them as huge, bandwidth-unfriendly, scanned documents.

The first are the lecture notes “Lectures on zeta functions and motives” by Yuri I. Manin of a course given in 1991.

One can download a scanned version of the paper from the homepage of Katia Consani as a huge 23.1 Mb file. Of F-un relevance is the first section “Absolute Motives?” in which

“…we describe a highly speculative picture of analogies between arithmetics over $\mathbb{F}_q $ and over $\mathbb{Z} $, cast in the language reminiscent of Grothendieck’s motives. We postulate the existence of a category with tensor product $\times $ whose objects correspond not only to the divisors of the Hasse-Weil zeta functions of schemes over $\mathbb{Z} $, but also to Kurokawa’s tensor divisors. This neatly leads to teh introduction of an “absolute Tate motive” $\mathbb{T} $, whose zeta function is $\frac{s-1}{2\pi} $, and whose zeroth power is “the absolute point” which is teh base for Kurokawa’s direct products. We add some speculations about the role of $\mathbb{T} $ in the “algebraic geometry over a one-element field”, and in clarifying the structure of the gamma factors at infinity.” (loc.cit. p 1-2)

I’d welcome links to material explaining this section to people knowing no motives.

The second one is the unpublished paper “Cohomology determinants and reciprocity laws : number field case” by Mikhail Kapranov and A. Smirnov.

This paper features in blog-posts at the Arcadian Functor, in John Baez’ Weekly Finds and in yesterday’s post at Noncommutative Geometry.

You can download every single page (of 15) as a separate file from here. But, in order to help spreading the Fun-gospel, I’ve made these scans into a single PDF-file which you can download as a 2.6 Mb PDF. In the introduction they say :

“First of all, it is an old idea to interpret combinatorics of finite sets as the $q \rightarrow 1 $ limit of linear algebra over the finite field $\mathbb{F}_q $. This had lead to frequent consideration of the folklore object $\mathbb{F}_1 $, the “field with one element”, whose vector spaces are just sets. One can postulate, of course, that $\mathbf{spec}(\mathbb{F}_1) $ is the absolute point, but the real problem is to develop non-trivial consequences of this point of view.”

They manage to deduce higher reciprocity laws in class field theory within the theory of $\mathbb{F}_1 $ and its field extensions $\mathbb{F}_{1^n} $. But first, let us explain how they define linear algebra over these absolute fields.

Here is a first principle : in doing linear algebra over these fields, there is no additive structure but only scalar multiplication by field elements. So, what are vector spaces over the field with one element? Well, as scalar multiplication with 1 is just the identity map, we have that a vector space is just a set. Linear maps are just set-maps and in particular, a linear isomorphism of a vector space onto itself is a permutation of the set. That is, linear algebra over $\mathbb{F}_1 $ is the same as combinatorics of (finite) sets.

A vector space over $\mathbb{F}_1 $ is just a set; the dimension of such a vector space is the cardinality of the set. The general linear group $GL_n(\mathbb{F}_1) $ is the symmetric group $S_n $, the identification via permutation matrices (having exactly one 1 in every row and column)

Some people prefer to view an $\mathbb{F}_1 $ vector space as a pointed set, the special element being the ‘origin’ $0 $ but as $\mathbb{F}_1 $ doesnt have a zero, there is also no zero-vector. Still, in later applications (such as defining exact sequences and quotient spaces) it is helpful to have an origin. So, let us denote for any set $S $ by $S^{\bullet} = S \cup { 0 } $. Clearly, linear maps between such ‘extended’ spaces must be maps of pointed sets, that is, sending $0 \rightarrow 0 $.

The field with one element $\mathbb{F}_1 $ has a field extension of degree n for any natural number n which we denote by $\mathbb{F}_{1^n} $ and using the above notation we will define this field as :

$\mathbb{F}_{1^n} = \mu_n^{\bullet} $ with $\mu_n $ the group of all n-th roots of unity. Note that if we choose a primitive n-th root $\epsilon_n $, then $\mu_n \simeq C_n $ is the cyclic group of order n.

Now what is a vector space over $\mathbb{F}_{1^n} $? Recall that we only demand units of the field to act by scalar multiplication, so each ‘vector’ $\vec{v} $ determines an n-set of linear dependent vectors $\epsilon_n^i \vec{v} $. In other words, any $\mathbb{F}_{1^n} $-vector space is of the form $V^{\bullet} $ with $V $ a set of which the group $\mu_n $ acts freely. Hence, $V $ has $N=d.n $ elements and there are exactly $d $ orbits for the action of $\mu_n $ by scalar multiplication. We call $d $ the dimension of the vectorspace and a basis consists in choosing one representant for every orbits. That is, $~B = { b_1,\ldots,b_d } $ is a basis if (and only if) $V = { \epsilon_n^j b_i~:~1 \leq i \leq d, 1 \leq j \leq n } $.

So, vectorspaces are free $\mu_n $-sets and hence linear maps $V^{\bullet} \rightarrow W^{\bullet} $ is a $\mu_n $-map $V \rightarrow W $. In particular, a linear isomorphism of $V $, that is an element of $GL_d(\mathbb{F}_{1^n}) $ is a $\mu_n $ bijection sending any basis element $b_i \rightarrow \epsilon_n^{j(i)} b_{\sigma(i)} $ for a permutation $\sigma \in S_d $.

An $\mathbb{F}_{1^n} $-vectorspace $V^{\bullet} $ is a free $\mu_n $-set $V $ of $N=n.d $ elements. The dimension $dim_{\mathbb{F}_{1^n}}(V^{\bullet}) = d $ and the general linear group $GL_d(\mathbb{F}_{1^n}) $ is the wreath product of $S_d $ with $\mu_n^{\times d} $, the identification as matrices with exactly one non-zero entry (being an n-th root of unity) in every row and every column.

This may appear as a rather sterile theory, so let us give an extremely important example, which will lead us to our second principle for developing absolute linear algebra.

Let $q=p^k $ be a prime power and let $\mathbb{F}_q $ be the finite field with $q $ elements. Assume that $q \cong 1~mod(n) $. It is well known that the group of units $\mathbb{F}_q^{\ast} $ is cyclic of order $q-1 $ so by the assumption we can identify $\mu_n $ with a subgroup of $\mathbb{F}_q^{\ast} $.

Then, $\mathbb{F}_q = (\mathbb{F}_q^{\ast})^{\bullet} $ is an $\mathbb{F}_{1^n} $-vectorspace of dimension $d=\frac{q-1}{n} $. In other words, $\mathbb{F}_q $ is an $\mathbb{F}_{1^n} $-algebra. But then, any ordinary $\mathbb{F}_q $-vectorspace of dimension $e $ becomes (via restriction of scalars) an $\mathbb{F}_{1^n} $-vector space of dimension $\frac{e(q-1)}{n} $.

Next time we will introduce more linear algebra definitions (including determinants, exact sequences, direct sums and tensor products) in the realm the absolute fields $\mathbb{F}_{1^n} $ and remarkt that we have to alter the known definitions as we can only use the scalar-multiplication. To guide us, we have the second principle : all traditional results of linear algebra over $\mathbb{F}_q $ must be recovered from the new definitions under the vector-space identification $\mathbb{F}_q = (\mathbb{F}_q^{\ast})^{\bullet} = \mathbb{F}_{1^n} $ when $n=q-1 $. (to be continued)

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the King’s problem on MUBs

MUBs (for Mutually Unbiased Bases) are quite popular at the moment. Kea is running a mini-series Mutual Unbias as is Carl Brannen. Further, the Perimeter Institute has a good website for its seminars where they offer streaming video (I like their MacromediaFlash format giving video and slides/blackboard shots simultaneously, in distinct windows) including a talk on MUBs (as well as an old talk by Wootters).

So what are MUBs to mathematicians? Recall that a d-state quantum system is just the vectorspace $\mathbb{C}^d $ equipped with the usual Hermitian inproduct $\vec{v}.\vec{w} = \sum \overline{v_i} w_i $. An observable $E $ is a choice of orthonormal basis ${ \vec{e_i} } $ consisting of eigenvectors of the self-adjoint matrix $E $. $E $ together with another observable $F $ (with orthonormal basis ${ \vec{f_j} } $) are said to be mutally unbiased if the norms of all inproducts $\vec{f_j}.\vec{e_i} $ are equal to $1/\sqrt{d} $. This definition extends to a collection of pairwise mutually unbiased observables. In a d-state quantum system there can be at most d+1 mutually unbiased bases and such a collection of observables is then called a MUB of the system. Using properties of finite fields one has shown that MUBs exists whenever d is a prime-power. On the other hand, existence of a MUB for d=6 still seems to be open…

The King’s Problem (( actually a misnomer, it’s more the poor physicists’ problem… )) is the following : A physicist is trapped on an island ruled by a mean
king who promises to set her free if she can give him the answer to the following puzzle. The
physicist is asked to prepare a d−state quantum system in any state of her choosing and give it
to the king, who measures one of several mutually unbiased observables on it. Following this, the physicist is allowed to make a control measurement
on the system, as well as any other systems it may have been coupled to in the preparation
phase. The king then reveals which observable he measured and the physicist is required
to predict correctly all the eigenvalues he found.

The Solution to the King’s problem in prime power dimension by P. K. Aravind, say for $d=p^k $, consists in taking a system of k object qupits (when $p=2l+1 $ one qupit is a spin l particle) which she will give to the King together with k ancilla qupits that she retains in her possession. These 2k qupits are diligently entangled and prepared is a well chosen state. The final step in finding a suitable state is the solution to a pure combinatorial problem :

She must use the numbers 1 to d to form $d^2 $ ordered sets of d+1 numbers each, with repetitions of numbers within a set allowed, such that any two sets have exactly one identical number in the same place in both. Here’s an example of 16 such strings for d=4 :

11432, 12341, 13214, 14123, 21324, 22413, 23142, 24231, 31243, 32134, 33421, 34312, 41111, 42222, 43333, 44444

Here again, finite fields are used in the solution. When $d=p^k $, identify the elements of $\mathbb{F}_{p^k} $ with the numbers from 1 to d in some fixed way. Then, the $d^2 $ of number-strings are found as follows : let $k_0,k_1 \in \mathbb{F}_{p^k} $ and take as the first 2 numbers the ones corresponding to these field-elements. The remaning d-2 numbers in the string are those corresponding to the field element $k_m $ (with $2 \leq m \leq d $) determined from $k_0,k_1 $ by the equation

$k_m = l_{m} * k_0+k_1 $

where $l_i $ is the field-element corresponding to the integer i ($l_1 $ corresponds to the zero element). It is easy to see that these $d^2 $ strings satisfy the conditions of the combinatorial problem. Indeed, any two of its digits determine $k_0,k_1 $ (and hence the whole string) as it follows from
$k_m = l_m k_0 + k_1 $ and $k_r = l_r k_0 + k_1 $ that $k_0 = \frac{k_m-k_r}{l_m-l_r} $.

In the special case when d=3 (that is, one spin 1 particle is given to the King), we recover the tetracode : the nine codewords

0000, 0+++, 0—, +0+-, ++-0, +-0+, -0-+, -+0-, –+0

encode the strings (with +=1,-=2,0=3)

3333, 3111, 3222, 1312, 1123, 1231, 2321, 2132, 2213

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