nLab Dieudonné determinant

Idea

The Dieudonné determinant is a variant of the notion of determinant which applies to matrices with values in division rings, such as the quaternions. It is crucial for making sense of the notion of “special linear group” over such coefficients, such as the group SL(2,H) over the quaternions.

Definition

Dieudonné 43 showed that for $K$ a division ring, there is a group homomorphism of the form

(1)$$\alpha \;\colon\; GL(n,K)/[GL(n,K), GL(n,K)] \longrightarrow K^\times/[K^\times, K^\times] \,,$$

where

• $GL(n,K)$ is the group of $n \times n$ invertible matrices with entries in $K$,

• $K^\times = GL(1,K)$ is the group of units of $K$, and

• $G/[G,G]$ is the abelianization of the group $G$.

Moreover, this group homomorphism (1) is uniquely determined by the following properties:

• If a row of the matrix $T$ is left multiplied by $a \in K^\times$ then $\alpha(T)$ is left multiplied by $[a] \in K^\times/[K^\times K^\times]$.

• If a multiple of one row of $T$ is added to another row of $T$, then $\alpha(T)$ is unchanged.

Composing the homomorphism $\alpha$ (1) with the quotient map

$$q \,\colon\, GL(n,K) \longrightarrow GL(n,K) / [GL(n,K), GL(n,K)]$$

gives the Dieudonné determinant

(2)$$det_{D} \,\coloneqq\, q \circ \alpha \;\colon\; GL(n,K) \longrightarrow K^\times / [K^\times, K^\times] \,.$$

The quaternionic case

In the case that $K$ is the ring of quaternions, $\mathbb{H}$, we have a group isomorphism

$$\mathbb{H}^\times / [ \mathbb{H}^\times, \mathbb{H}^\times ] \longrightarrow \mathbb{R}_{\gt 0}$$

where $\mathbb{R}_{\gt 0}$ is made into a group using multiplication of real numbers, coming from the fact that now the commutator subgroup $[\mathbb{H}^\times, \mathbb{H}^\times]$ is the group Sp(1) of quaternions $q$ with $|q| = 1$, and that any invertible quaternion can uniquely be written as $a q$ where $a \in \mathbb{R}_{\gt 0}$ and $\left\vert q \right\vert = 1$.

Thus, in this case we can write the Dieudonné determinant (2) more concretely as a group homomorphism of the form

$$det_{D} \,\colon\, GL(n,\mathbb{H}) \longrightarrow \mathbb{R}_{\gt 0} \,.$$

Furthermore, for any $n \times n$ quaternionic matrix $T$ that is not invertible, setting $det_D(T) \coloneqq 0$ extends the Dieudonné determinant to a function of the form

(3)$$det_{D} \;\colon\; M_n(\mathbb{H}) \longrightarrow \mathbb{R}_{\ge 0} \,,$$

where $\mathrm{M}_n(\mathbb{H})$ is now the full matrix algebra of $n \times n$ matrices with quaternion entries. This function (3) is a monoid homomorphism, in that:

(4)$$det_{D}(S T) \,=\, det_{D}(S) det_{D}(T), \qquad det_{D}(1) = 1 \,.$$

Relation to the Study determinant

There is another notion of determinant for quaternionic matrices, the Study determinant. This turns out to be just the square of the Dieudonné determinant, but it often gives a more convenient way to compute the Dieudonné determinant.

Any matrix $T \in \mathrm{M}_n(\mathbb{H})$ determines by right multiplication a homomorphism of left $\mathbb{H}$-modules $T \colon \mathbb{H}^n \to \mathbb{H}^n$. Choosing any element $i \in \mathbb{H}$ with $i^2 = -1$ gives $\mathbb{H}^n$ the structure of a left $\mathbb{C}$-module: indeed, a complex vector space of dimension $2n$. In this way we can identify $T \colon \mathbb{H}^n \to \mathbb{H}^n$ with a complex-linear transformation of a complex vector space, and define its determinant in the usual way for such a transformation.

This determinant turns out not to depend on the choice of $i \in \mathbb{H}$ with $i^2 = -1$, and it is called the Study determinant $det_S(T)$.

It clearly obeys

$$det_S(S T) = det_S(S) det_S(T), \qquad det_S(1) = 1 \,.$$

A priori it is complex-valued, but in fact it takes nonnegative real values, because one can show (Arlaksen 96) that

$$det_{S}(T) = det_{D}(T)^2 \,,$$

where the determinant on the right is the Dieudonné determinant (3).

Thus, by (4), the Study determinant on $n \times n$ quaternionic matrices defines a monoid homomorphism

$$det_{S} \;\colon\; M_n(\mathbb{H}) \longrightarrow \mathbb{R}_{\ge 0}$$

which contains exactly as much information as the Dieudonné determinant.

Examples and applications

Over any division ring, the Dieudonné determinant of an invertible $2 \times 2$ matrix has the following form:

$$\det \left({\begin{array}{cc} a & b \\ c & d \end{array}}\right) = \left\lbrace{\begin{array}{ccc} [-c b] & \text{if } \; a = 0 \\ [a d - a c a^{-1}b] & \text{if } \; a \ne 0 \end{array}}\right. \, .$$

Here $[x]$ stands for the equivalence class of $x \in K^\times$ in the abelianization $K^\times/[K^\times, K^\times]$.

The special linear group $\mathrm{SL}(n,\mathbb{H})$ over the quaternions (e.g. SL(2,H)) is defined to be the group of $n \times n$ quaternionic matrices for which the Dieudonné determinant equals 1, or equivalently for which the Study determinant equals 1.

We can show that the quaternionic unitary group Sp(n) is a subgroup of $\mathrm{SL}(n,\mathbb{H})$, as follows. So, unlike the real and complex cases, there is no additional concept of “special unitary group” in the quaternionic case.

Every quaternionic unitary matrix is conjugate to a diagonal one since every element is conjugate to one in a maximal torus, $\mathrm{U}(n)$ contains a maximal torus of $\mathrm{Sp}(n)$, and every matrix in $\mathrm{U}(n)$ is conjugate to a diagonal one. Thus, every $g \in Sp(n)$ is conjugate to one of the form $\mathrm{diag}(q_1, \dots, q_n)$ with $q_i \in \mathbb{C} \subset \mathbb{H}$ and $|q_i| = 1$ for all $i$. By the defining properties of the Dieudonné determinant we have $det_D(\mathrm{diag}(q_1, \dots, q_n)) = 1$, and since this determinant is invariant under conjugation we have $det_D(g) = 1$ for all $g \in Sp(n)$.

Draxl’s pre-determinant

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 $\delta\epsilon\tau(T)$, introduced by Draxl, 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))

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 inferred that the Dieudonné predeterminant can be generically presented as a signed product of quasideterminants.

References

The concept is due to

• Jean Dieudonné, Les déterminants sur un corps non commutatif, Bulletin de la Société Mathématique de France, Volume 71 (1943), p. 27-45 (numdam:BSMF_1943__71__27_0)

Exposition and review:

• Helmer Aslaksen, Quaternionic determinants, The Mathematical Intelligencer 18, 57–65 (1996) (doi:10.1007/BF03024312, pdf)

• Nir Cohen, Stefano De Leo, The quaternionic determinant, El. J. Lin. Alg. 7, 100-111 (2000) (arXiv:math-ph/9907015)

• Nigel Hitchin, $SL(2)$ over the octonions (arXiv:1805.02224)

• Joás Venâncio, Carlos Batista, Section 2 of: Two-Component Spinorial Formalism using Quaternions for Six-dimensional Spacetimes (arXiv:2007.04296)

See also:

• Wikipedia, Dieudonné determinant

Comparison to quasideterminants is in

• Israel Gelfand, Vladimir Retakh, Robert Lee Wilson, Quaternionic quasideterminants and determinants, 2002 (math.QA/0206211)

• Peter Draxl, Section III.20 of: Skew fields, London Math. Soc. Lecture notes 81, Cambridge University Press 1983 (doi:10.1017/CBO9780511661907, ISBN:9780511661907)

For lectures on quasideterminants see

• V. Retakh, R. Wilson, Advanced course on quasideterminants and universal localization, 124 pp, CRM, Barcelona, 2007 (pdf)