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added pointer to this readable review:
added pointer to
Thanks. I have expanded the Idea-section a tad and touched formatting, hyperlinking, cross-linking, wording, and punctuation in sections 2 and 3.
In particular, I gave the notation “$det$” a subscript: “$det_{Dieu}$” (since it does not reduce to the usual $det$ on complex matrices); and where it said “if two rows are exchanged” I inserted the word “consecutive”, since I guess this is needed here.
(since it does not reduce to the usual detdet on complex matrices)
How so? For C we have C^⨯/[C^⨯,C^⨯] = C^⨯, so the determinant does reduce to the usual determinant.
True. But the quaternionic Dieudonné determinant does not reduce to the complex determinant along the inclusion of complex matrices into quaternionic matrices (but to the absolute value of the complex determinant).
So I suppose the plain notation $det$ is consistent in itself. But it should help to have the subscript around to avoid the evident pitfall. Best would be to add a paragraph on this point…
I see. The notation I would use would be det_C and det_H, in complete analogy to how we denote the base field for tensor products, etc.: ⊗_C, ⊗_H.
That sounds good to me. Let’s make it “$det_K$” in the entry.
While we are at streamlining the notation, here are two more changes I would be inclined to make:
change “$K$” to “$\mathbb{K}$ (or to “$k$”, but better “$\mathbb{K}$”)
change “$sdet$” to something that looks less like the standard notation for super-determinants. If we do/did keep “$det_{Dieu}$” then the evident choice here would be “$det_{Stud}$”, but if/since we don’t, this is not the evident choice. It’s still a possible choice, though.
Draxl introduces a more primitive notion, Dieudonné predeterminant. Given a generic matrix $T$ over a skew-field one performs its Gauss decomposition in the form $T = U D L$ where $U$ is upper triangular unidiagonal, $D$ diagonal and $L$ lower triangular matrix, Dieudonné predeterminant $\delta\epsilon\tau(T)$ is the product of the entries of the diagonal part $D$ upside down. If the matrix is not generic, question of rank surface out and the appropriate Bruhat decomposition should be chosen instead. Dieudonné determinant is the image of $\delta\epsilon\tau(T)$ under the projection to the Abeliazation. It is well known that the Gauss decomposition of matrices over a noncommutative ring has an expression in terms of quasideterminants, as shown by Gelfand and Retakh in their foundational papers around 1990.
I also added references to Draxl and a later paper of Gelfand at al. on quasi and other determinants for quaternions.
For matrices over noncommutative rings, quasideterminants are much more fundamental notion than the Dieudonné determinant. Not only Berezinians, Dieudonné (pre)determinant, quantum determinants, inverse matrices and Gauss decomposition but also things like noncommutative analogues of Hankel determinants, noncommutative symmetric functions, quasisymmetric functions, continued fractions, and other important notions have their simplest expressions in terms of quasideterminants, sometimes even simpler than commutative analogues are expressed in terms of determinants (e.g Sylvester laws).
What is the tilde sign above the arrow in formula (1) in the entry Dieudonné determinant ? Is it an iso ?
Removed the $\sim$ from (1), since it used to be claimed as an isomorphism, but it turns out not to be so in the case $K=\mathbb{F}_2$.
Also, Oscar Cunningham seems to have a convincing argument that this is the only counterexample among division rings (it comes down to needing some $\lambda$ such that $\lambda$ and $1-\lambda$ are units in a the division ring). In the case of $\mathbb{F}_2$, the abelianisation is iso to $\mathbb{Z}/2$, but the target of the determinant on the invertible matrices is just $\{1\}$.
Zoran,
on your rev 12:
I have looked up and tried to add pointer to where Draxl says these things in his book.
Beyond that, I have touched the paragraph on the Draxl pre-determinant a little (here):
Where you had “generic” I added “i.e.: invertible”.
After the decomposition I added “This is called the strict Bruhat normal form” with pointer to Draxl’s definition.
I am not sure what to make of your “upside down” in “the entries of the diagonal part $D$ upside down” (?)
Similarly in the next line, I am not sure what to make of “surface out” in “If the matrix is not generic, question of rank surface out ” (?) Maybe you mean “the question arises”? But even then, isn’t non-full rank essentially the same a not being generic? Maybe you could clarify here.
Then the last line of the paragraph – “It is well known that the Gauss decomposition of matrices over a noncommutative ring has an expression in terms of quasideterminants” – is lacking some logical tissue with the previous material. Maybe this sentence could continue with “…Draxl’s pre-determinant is such a quasi-determinant” (?).
If the matrix is not generic, question of rank surface out
I took this to mean “If the matrix is not generic, questions of rank surface.”
By the way, while rev 10 kept mon Dieu, just changing it to D, I still think that Dmitri in #10 had the right idea:
In the end, writing “$det_D$” throughout is as ambiguous as writing “$det$” throughout: both notations fail to make their dependence on the choice of $K$ explicit.
And since the distinction between “$det_D$” and “$det_S$” is of interest only for the history of the subject, not for the mathematical content itself, I vote for writing, in our entry, “$det_K$” for the former and “$det_K^2$” for the latter.
(Or rather, I really vote for writing $det_{\mathbb{K}}$ for the former and $det^2_{\mathbb{K}}$ for the latter…)
Sorry for a rather hasty and temporarily a bit misleading contribution.
Draxl determinant is not a quasideterminant, but a signed product of $n$ quasideterminants (just like the determinant of a commutative matrix, quantum determinant and Berezinian are).
Quasideterminants are noncommutative rational functions rather than polynomial and they appear more fundamental in this generality; when some sort of a polynomial determinant makes sense it is usually a simple expression (usually a product) of quasideterminants. For each entry of the matrix there is one quasideterminant, $|A|_{ij}$ (the ones in the same row or the same column are related by a ratio of lower size quasideterminants, this is called a homological relation). Now if you take a matrix and its lower right corner minor matrices of all sizes from one to 1 to n (these are sometimes called principal) and you take the quasideterminants with the left upper corner distinguished for each of them these are the principal quasiminors. So if you take a product of all $n$ principal quasiminors (no summation involved, just a simple product) you get the determinant in commutative case. Conversely, if you take the determinant of a matrix and divide by the determinant of its principal minor of size $(n-1)\times(n-1)$, then the ratio is the quasideterminant (in commutative case).
Upside down, pardon my English, I meant in order from upper to lower.
Maybe you mean “the question arises”
Yes, but I was too vague here.
As far as generic versus invertible/rank, Draxl assumes maximal row rank anyway, but I meant belonging to the big Bruhat cell, intuitively dense/generic (it is a specialization of a Cohn localization of a free algebra, which is in more specific sense dense in free algebra on $n^2$ entries). Of course, the big Bruhat cell can be viewed beyond invertible matrices, but I think one should be a bit careful there. Now, other Bruhat cells have in commutative case certain codimension, so my allusion to rank was in this sense. Thus I said “questions of rank” as a type of reasoning. This was I see misleading. I am grading right now so I have no time to refresh my memory in sources (and my own research on related notions in Cohn localization and consequences of Gauss decomposition) before Sunday, but I should write this more clearly eventually.
I also meant paranthetically that in some more general ring generalizations one can not describe in Draxl terms, but needs some machinery based on evaluation procedures of elements from the algebra of abstract noncommutative rational functions and certain combinatorial heights, genericity and other combinatorial questions in abstract algebra may be used. You see, abstract nc rational functions may or may not evaluate at specific elements of a specific ring; more importantly there is more than one expression for a rational function and one of the equivalent expressions works for some elements and another expression works for another elements. Generically you do not need to make a choice.
Erased (duplicate).
This is probably more clear.
Draxl 83 introduces a more primitive notion, the Dieudonné predeterminant:
Given an invertible matrix $T$ over a skew-field (and in some other cases) there is a strict Bruhat normal form of $T$ (Draxl 83, Sec. 19, Thm. 1, Def. 1 (p. 128)) $T = U D P L$ where
$P$ is a permutation matrix
$U$ is upper triangular unidiagonal,
$D$ diagonal
$L$ lower triangular unidiagonal matrix.
The case when for $P$ the identity matrix can be taken may be viewed as generic as such matrices are dense in a number of meanings and contexts. This is the case of belonging to the big Bruhat cell or equivalently to the main Gauss cell, and the decomposition $T = U D L$ is the Gauss decomposition. Recall that other Bruhat cells are in commutative case of higher codimension, hence not dense, and similar statements can be made in a number of noncommutative contexts. Shifted Gauss cells, which correspond to a decomposition of matrices in the form $P U D L$ for $P$ fixed are also dense in the same sense as they are simply the shifts (by multiplication by an invertible matrix $P$) of the main cell.
The Dieudonné predeterminant, introduced by Draxl, $\delta\epsilon\tau(T)$ is the product of the entries of the diagonal part $D$ upside down (Draxl 83, Sec. 20, Def. 1 (p. 133)) if the matrix is invertible and zero otherwise.
The Dieudonné determinant is then the image of $\delta\epsilon\tau(T)$ under the projection to the abelianization (Draxl 83, Sec. 20, Cor. 1 (p. 135))
This is probably more clear.
Draxl 83 introduces a more primitive notion, the Dieudonné predeterminant:
Given an invertible matrix $T$ over a skew-field (and in some other cases) there is a strict Bruhat normal form of $T$ (Draxl 83, Sec. 19, Thm. 1, Def. 1 (p. 128)) $T = U D P L$ where
$P$ is a permutation matrix
$U$ is upper triangular unidiagonal,
$D$ diagonal
$L$ lower triangular unidiagonal matrix.
The case when for $P$ the identity matrix can be taken may be viewed as generic as such matrices are dense in a number of meanings and contexts. This is the case of belonging to the big Bruhat cell or equivalently to the main Gauss cell, and the decomposition $T = U D L$ is the Gauss decomposition. Recall that other Bruhat cells are in commutative case of higher codimension, hence not dense, and similar statements can be made in a number of noncommutative contexts. Shifted Gauss cells, which correspond to a decomposition of matrices in the form $P U D L$ for $P$ fixed are also dense in the same sense as they are simply the shifts (by multiplication by an invertible matrix $P$) of the main cell.
The Dieudonné predeterminant, introduced by Draxl, $\delta\epsilon\tau(T)$ is the product of the entries of the diagonal part $D$ upside down (Draxl 83, Sec. 20, Def. 1 (p. 133)) if the matrix is invertible and zero otherwise.
The Dieudonné determinant is then the image of $\delta\epsilon\tau(T)$ under the projection to the abelianization (Draxl 83, Sec. 20, Cor. 1 (p. 135))
Urs said
Then the last line of the paragraph – “It is well known that the Gauss decomposition of matrices over a noncommutative ring has an expression in terms of quasideterminants” – is lacking some logical tissue with the previous material. Maybe this sentence could continue with “…Draxl’s pre-determinant is such a quasi-determinant” (?).
Is the following better ?
It is well known that the Gauss decomposition of matrices over a noncommutative ring has a simple expression in terms of quasideterminants, as shown by Gelfand-Retakh 02 (and in their earlier references, around 1990), from which it can be infered that the Dieudonné predeterminant can be generically presented as a signed product of quasideterminants.
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