Exterior algebra

Posted on 2026/01/18 by arkajad

Geometric algebra was not born in an ivory tower; it grew out of three very human lives, each marked by faith, doubt, and an almost mystical hunger for structure.
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The quiet mystic: Grassmann
Hermann Grassmann was a provincial schoolteacher, a Lutheran theologian manqué who raised a large family and spent his evenings inventing an algebra for space that nobody wanted to read.
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Between lessons, political essays, Freemason meetings, and plans to evangelize China, he wrote the Ausdehnungslehre—a book that looks more like a private revelation than a textbook, weaving geometry, language, and scripture into one symbolic universe.​

The tormented romantic: Hamilton
William Rowan Hamilton lived in an observatory near Dublin, surrounded by telescopes, poetry, and unfulfilled love affairs.
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He discovered quaternions in a flash of illumination on a bridge, carved the fundamental formula into the stone, and then spent years struggling with alcohol, domestic tensions, and a restless, almost alchemical desire to turn geometry into pure algebraic light.
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The ethical freethinker: Clifford
William Kingdon Clifford was a brilliant, fragile Victorian who burned himself out by his mid‑thirties, trying to fuse mathematics, physics, and a new secular ethics.
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He admired Grassmann, extended his ideas, and at the same time attacked traditional religion, arguing that even belief itself must obey strict moral laws—no faith without evidence, no comfort without intellectual honesty.
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The private life of geometric algebra
Seen together, Grassmann, Hamilton, and Clifford form something like a hidden trinity of geometric algebra: the quiet mystic of extension, the romantic discoverer of hypercomplex numbers, and the ethical revolutionary who welded them into a single geometric language.
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Their theories are usually presented as cold formalism, yet they were written under the pressure of illness, unrequited love, religious struggle, and a deep, almost esoteric conviction that space itself carries meaning.
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Toward a mathematics of consciousness?
If geometric algebra unifies length, angle, and orientation into one symbolic field, it is tempting to imagine a future “extension theory” in which thoughts, qualia, and inner states are modeled as multivectors in a higher‑dimensional cognitive space.
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A bold extrapolation is that consciousness might eventually be described as a kind of global geometric field over the brain and its environment, with attention, memory, and emotion appearing as special “directions” and “blades” in an unseen algebraic universe.
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In such a picture, Grassmann’s extension, Hamilton’s quaternions, and Clifford’s geometric product would be early pages in a much larger Ausdehnungslehre der Seele—an algebra of mind where the geometry of space and the geometry of experience finally meet.

In this post we describe the exterior algebra, invented by Hermann Grassmann around 1844.

In “Tensors are geometric objects” and “`Tensors on a picnic”} we have defined contravariant, covariant, and mixed tensors, by their transformation properties under a change of a basis. Let V be an n-dimensional real vector space. The space of p-contravariant tensors is usually denoted \bigotimes^p V. The space of p-covariant tensors is denoted \bigotimes^p V^*. It can be interpreted in two ways, either as (\bigotimes^p V)^*, or as \bigotimes^p (V^*). Both are correct, since if t^* is p times covariant, we can interpret t^* as a linear form on p-times contravariant tensors by

    \[ <t^*,t>={t^*}_{i_1\ldots i_p}t^{i_1\ldots i_p},\]

and every linear form on \bigotimes^p V can be uniquely represented in this way.

In the literature we can find different notations, and it is good to know about them. I will partly employ the notation used by Marian Fecko in his book [1]. I truly love this book! Here is my review.
In this book T^p(V) is used for \bigotimes^p V, T_q(V) for \bigotimes^q V^*, T^p_q(V) for the space of p-contravarian and q-covariant tensors. They are called tensors of type \binom{p}{q}. Tensors of rype \binom{0}{p} that are completely antisymmetric are called p-forms.

If t_1,t_2,\ldots,t_p are in V, we denote by t_1\otimes t_2\otimes\cdots\otimes t_p the tensor t with components

    \[t^{i_1i_2\ldots i_p}=t_1^{i_1}t_2^{i_2}\cdots t_p^{i_p}.\]

Tensors of this form are called simple. Simple tensors span all of \bigotimes^p V. The same applies to covariant tensors.

The most important is the universal property of \bigotimes^p V: Every p-linear map f from V\times\cdots V (p times) to a vector space F extends to a unique linear map (denoted by the same letter f) from \bigotimes^p V to F, such that

    \[f(v_1,\ldots, v_p)= f(v_1\otimes\cdots\otimes v_p).\]

If e_i, (i=1,\ldots n) is a basis in V, then e_{i_1}\otimes\cdots\otimes e_{i_p} is a basis in \bigotimes^p V. Tensor indices refer to such a basis: if t\in\bigotimes^p V, then

    \[ t=t^{i_1\ldots i_p}e_{i_1}\otimes\cdots\otimes e_{i_p}.\]

Of special interest are (totally) antisymmetric (or “skew-symmetric”) tensors. These are tensors, covariant or contravariant) that change sign under odd permutations of their indices. Let \mathfrak{S}_p denote the permutation group in p elements. Then \mathfrak{S}_p acts on \bigotimes^p V by

    \[ \sigma\cdot \left(t^{i_1}\otimes\cdots\otimes t^{i_p}\right) = t^{\sigma^{-1}(i_1)}\otimes\cdots\otimes t^{\sigma^{-1}(i_1)},\quad\sigma\in\mathfrak{S}_p.\]

This is for simple tensors. For general tensors we have simply the permutation of indices. A tensor t in \bigotimes^p V is {\it antisymmetric} if, for every \sigma\in \mathfrak{S}_p,

    \[\sigma\cdot t=\epsilon_\sigma\, t,\]

where \epsilon_\sigma is the sign of \sigma, \epsilon_\sigma=+1 for \sigma even, \epsilon_\sigma = -1 if \sigma is odd.

We then define the antisymmetrization operator:

    \[ {\bf a}\cdot t=\sum_{\sigma\in \mathfrak{S}_p}\epsilon_\sigma\, (\sigma\cdot t).\]

We can also characterize antisymmetric tensors using generalized Kronecker deltas discussed in “It’s all about permutations“. Namely, t is antisymmetric if and only if

(0)   \[ \delta^{i_1\ldots i_p}_{j_1\ldots j_p}\,t^{j_1\ldots j_p}=p!\,t^{i_1\ldots i_p}.\]

The antisymmetrization operator can be then also written in components as

    \[ ({\bf a}\cdot t)^{i_1\ldots i_p}=\delta^{i_1\ldots i_p}_{j_1\ldots j_p}\,t^{j_1\ldots j_p}.\]

Exercise 1. Justify the two last statements.

There is also a variation of the antisymmetrization operator, denoted by \pi^A in [1]:

    \[ \pi^A(t)=\frac{1}{p!}{\bf a}\cdot t\]

for t\in T_p(V), and with {\bf a}\cdot operator defined above. It has the advantage that \pi^A\circ \pi^A=\pi^A. Then, for t\in T_p(V) we write

    \[ t_{[a\ldots b]}=\pi^A(t)_{a\ldots b}.\]

The same for t\in T^p(V).

The p–th exterior power \Lambda^p V

The vector space of p-fold antisymmetric covariant (resp. contravariant) tensors is denoted \Lambda^p V* (resp. \Lambda^p V), and is called p-th exterior power of V* – the space of p-forms (resp. p-vectors). Notice that for p>n the only antisymmetric tensor is the zero tensor. (Why?). The whole exterior algebra \Lambda V^* is then defined as a direct sum:

    \[ \Lambda V^*={\Lambda}^0 V^*\oplus{\Lambda}^1V^*\oplus\cdots\oplus\Lambda^p V^*,\]

where {\Lambda}^0 V^* is understood as the field \bR itself.
If e_i,\quad (i=1,\ldots,n) is a basis in V, and e^i is the dual basis in V^*, then

    \[e_{i_1\ldots i_p}=\delta^{j_1\ldots j_p}_{i_1\ldots i_p}e_{j_1}\otimes\cdots\otimes e_{j_p},\]

i_1<i_2<\cdots <i_p is a basis in {\Lambda}^p V, and

    \[e^{i_1\ldots i_p}=\delta_{j_1\ldots j_p}^{i_1\ldots i_p}e^{j_1}\otimes\cdots\otimes e^{j_p},\]

i_1<i_2<\cdots <i_p is a basis in {\Lambda}^p V^*.

The wedge product

We now define the algebra product in \Lambda V. For t\in{\Lambda}^p V and s\in{\Lambda}^q V we define t\wedge s \in {\Lambda}^{p+q} V by

    \[ t\wedge s=\frac{1}{p!q!}{\bf a}\cdot (t\otimes s),\]

or explicitly, in components:

(1)   \[ (t\wedge s)^{i_1\ldots i_pi_{p+1}\ldots i_{p+q}}=\frac{1}{p!q!} \delta^{i_1\ldots i_pi_{p+1}\ldots i_{p+q}}_{j_1\ldots j_pj_{p+1}\ldots j_{p+q}}\,t^{j_1\ldots j_p}\,s^{j_{p+1}\ldots j_{p+q}}.\]

The wedge product is then extended to the whole \Lambda V by linearity. Using the properties of Kronecker deltas it can be then shown that the product is associative, so that \Lambda V becomes and associative algebra (with unit 1) – the exterior (or Grassmann) algebra of V. To prove associativity we will need a little lemma. In fact it belongs to the post on Kronecker deltas, but while writing that post I forgot about this useful property while writing about them, so here it is:

Lemma 1. For any totally antisymmetric array t^{j_1\ldots j_r} we have

    \[\delta^{i_1\ldots i_r}_{j_1\ldots j_r}t^{j_1\ldots j_r}=r!\,t^{i_1\ldots i_r}.\]

The proof follows directly by applying Eq. (3) from the previous post and using antisymmetry of t – we get r! identical terms.

Proof of associativity.

For t\in\Lambda^p V^*, s\in \Lambda^q V^*, u\in\Lambda^r V^*, using (adapted) Eq. (1) we have

    \[ (t\wedge s)_{j_1\ldots j_pj_{p+1}\ldots j_{p+q}}=\frac{1}{p!q!} \delta_{j_1\ldots j_pj_{p+1}\ldots j_{p+q}}^{k_1\ldots k_pk_{p+1}\ldots k_{p+q}}\,t_{k_1\ldots p_p}\,s_{k_{p+1}\ldots k_{p+q}},\]

and

    \[ \begin{split}&((t\wedge s)\wedge u)_{i_1\ldots i_{p+q+r}}\\ &= \frac{1}{(p+q)!r!}\,\delta_{i_1\ldots i_{p+q+r}}^{j_1\ldots j_{p+q}j_{p+q+1}\ldots j_{p+q+r}}(t\wedge s)_{j_1\ldots j_{p+q}} u_{j_{p+q+1}\ldots j_{p+q+r}}\\ &=\frac{1}{(p+q)!p!q!r!}\delta_{i_1\ldots i_{p+q+r}}^{j_1\ldots j_{p+q+r}}\delta_{j_1\ldots j_{p+q}}^{k_1\ldots k_{p+q}}\,t_{k_1\ldots p_p}\,s_{k_{p+1}\ldots k_{p+q}}u_{j_{p+q+1}\ldots j_{p+q+r}}\\ &=\frac{1}{p!q!r!} \delta_{i_1\ldots i_{p+q+r}}^{k_1\ldots k_{p+q}j_{p+q+1}\ldots j_{p+q+r}}\,t_{k_1\ldots p_p}\,s_{k_{p+1}\ldots k_{p+q}}u_{j_{p+q+1}\ldots j_{p+q+r}}, \end{split}\]

where the last equality is obtained using Eq. (0), and the fact that \delta_{i_1\ldots i_{p+q+r}}^{j_1\ldots j_{p+q+r}} is antisymmetric in indices j_1,\ldots, j_{p+q}.
Calculating now, in a similar way, t\wedge (s\wedge u we obtain, as it is easy to guess, the same result. This proves the associativity of the wedge product.\qed

Remark The above definition (1) of the wedge product works fine for \bR and \bC, but may fail for a general field, when the factor 1/(p!q!) may cause the problem (for instance in the field of integers modulo 2 we have -1=+1 and 2!=0). Therefore mathematicians that like to be as general as possible, introduce the exterior algebra (of a “module over a commutative ring”) differently.

References
[1] Fecko, M., “Differential Geometry and Lie Groups for Physicists”, Cambridge University Press 2006.

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5 Responses to Exterior algebra

  1. Anna says:
    2026/01/19 at 16:05

    Wow, what a densely packed post – from both a historical and mathematical viewpoints.

    Found a few typos, with fonts:
    (1) Tensors of this form are called {\it simple}
    (2) ({\bf Why?}).

    and
    (3) in formula after the phrase
    “The antisymmetrization operator can be then also written in components as”
    first j should be j_1, not j^1

    (4) This is not a typo, but I don’t know what is “a.t” in the formula below the phrase:
    “There is also a variation of the antisymmetrization operator, denoted by \pi^A in [1]”

  2. Anna says:
    2026/01/19 at 16:24

    Ark, may I ask – why formula for simple tensors is more complicated than for general ones? For them we have simply the permutation of indices, while for simple tensors we have inverse sigmas. Is it similar to the situation when coordinates of vectors transform using the inverse matrix? Or when selecting a gauge: b or b’, the corresponding gauge transformation is x’ = rho(g^{-1}) x

    • arkajad says:
      2026/01/19 at 17:08

      Thanks for corrections. Fixed. And your guess is correct. Permuting vectors is a kind of reverse to permuting coordinates. You can see it by choosing basis vectors.

  3. Anna says:
    2026/01/19 at 17:40

    As regards Exercise 1, it is just the formula 16.5-4 from Korn & Korn. Since we should not be confiding, I’m thinking how to show this rigorously. Perhaps, one can use formula from the Afternotes to https://arkadiusz-jadczyk.eu/blog/2026/01/kronecker-generalized-deltas-and-levi-civita-epsilons/?
    By the way, if we take s=r in the general formula for splitting off deltas, it seems that we should get
    delta_{j1,…jr}^{i1,…,ir} = n!/(n-r)!
    But according to the textbook, this is right only if i_n = j_n. And what about the general case when s=r?

  4. Anna says:
    2026/01/20 at 20:19

    In the “Remark The above definition (1) of the wedge product works fine for \bR and \bC”, the bold letters R and C are invisible in the text on site.

    On closer inspection, Exercise 1 is essentially the same as Lemma 1. Formula (3) from the “generalized Kronecker deltas” explains it all, doesn’t it?

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