01.02.19 · foundations / groups

Tensor algebra, exterior algebra, symmetric algebra

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Anchor (Master): Grassmann 1844 Ausdehnungslehre; Cartan 1922 Lecons; Bourbaki Algebre Ch. III

Intuition [Beginner]

Start with a vector space $V$ — a collection of vectors you can add and scale. The tensor algebra builds every possible product of these vectors: products of one vector, products of two, products of three, and so on. It is the most general algebra you can build from $V$ , like writing polynomials where the variables are your basis vectors. Every other way to multiply vectors is a special case of this one.

Two special cases stand out. The exterior algebra keeps only those products that change sign when you swap two factors — like how a signed area flips when you reverse orientation. This algebra measures areas, volumes, and their higher-dimensional analogues. The symmetric algebra keeps only products that stay the same under swapping — like ordinary multiplication of numbers, where the order does not matter. This algebra produces polynomial expressions.

Why does this concept exist? The tensor algebra is the universal framework for multiplying vectors. The exterior algebra captures oriented areas and volumes. The symmetric algebra captures polynomial expressions. Together they organise every way to build multi-variable expressions from a single vector space.

Visual [Beginner]

A tower of blocks stacked vertically. The bottom block represents scalars (dimension 1). The next block represents $V$ itself (dimension $n$ ). Each successive block represents products of one more vector, with dimension multiplied by $n$ each time. Two arrows branch from the tower: one pointing left to a shorter tower labelled "exterior algebra" and one pointing right to a widening tower labelled "symmetric algebra."

The exterior algebra reaches a peak at the middle level and then shrinks back to zero. The symmetric algebra grows without bound.

Worked example [Beginner]

Consider $V = R^{2}$ , a 2-dimensional space with basis vectors $e_{1}$ and $e_{2}$ .

Step 1. The tensor algebra builds a tower of products at increasing lengths. Length 0 is just the scalars: dimension 1. Length 1 is $V$ itself: dimension 2. Length 2 gives all pairwise products: dimension $2 \times 2 = 4$ . Length 3 gives all products of three vectors: dimension $2 \times 2 \times 2 = 8$ .

Step 2. The exterior algebra at each level keeps only anti-symmetric products. At length 0, dimension 1. At length 1, dimension 2 (same as $V$ ). At length 2, swapping the two factors changes the sign, so the only independent anti-symmetric product combines $e_{1}$ and $e_{2}$ : dimension 1. At length 3 and above, you cannot form an anti-symmetric product from only 2 basis vectors: dimension 0.

Step 3. The symmetric algebra at each level keeps only symmetric products. At length 0, dimension 1. At length 1, dimension 2. At length 2, the products $e_{1}$ with $e_{1}$ , $e_{1}$ with $e_{2}$ , and $e_{2}$ with $e_{2}$ are all distinct: dimension 3. At length 3, the products are $e_{1}$ with $e_{1}$ with $e_{1}$ , $e_{1}$ with $e_{1}$ with $e_{2}$ , $e_{1}$ with $e_{2}$ with $e_{2}$ , and $e_{2}$ with $e_{2}$ with $e_{2}$ : dimension 4.

What this tells us: the exterior algebra of $R^{2}$ has total dimension $1 + 2 + 1 = 4$ (which is $2^{2}$ ), while the symmetric algebra grows without bound.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $K$ be a field and let $V$ be a finite-dimensional vector space over $K$ ^{[vector space]}. The $k$ -th tensor power of $V$ is $V^{\otimes k} = k times V \otimes_{K} V \otimes_{K} \dots \otimes_{K} V$ with the convention $V^{\otimes 0} = K$ ^{[tensor product]}.

Definition (Tensor algebra). The tensor algebra of $V$ is the graded $K$ -algebra $T (V) = k \geq 0 ⨁ V^{\otimes k}$ with multiplication given by concatenation: $(v_{1} \otimes \dots \otimes v_{k}) \cdot (w_{1} \otimes \dots \otimes w_{l}) = v_{1} \otimes \dots \otimes v_{k} \otimes w_{1} \otimes \dots \otimes w_{l}$ .

Definition (Exterior algebra). The exterior algebra (or Grassmann algebra) of $V$ is the quotient $Λ (V) = T (V) /, I_{ext}$ where $I_{ext}$ is the two-sided ideal of $T (V)$ generated by all elements $v \otimes v$ for $v \in V$ . The image of $v_{1} \otimes \dots \otimes v_{k}$ in $Λ (V)$ is written $v_{1} \land \dots \land v_{k}$ and called the wedge product. The grading descends: $Λ (V) = ⨁_{k \geq 0} Λ^{k} (V)$ where $Λ^{k} (V) = V^{\otimes k} / (I_{ext} \cap V^{\otimes k})$ ^{[Bourbaki Algebre Ch. III]}.

Definition (Symmetric algebra). The symmetric algebra of $V$ is the quotient $Sym (V) = T (V) /, I_{sym}$ where $I_{sym}$ is the two-sided ideal of $T (V)$ generated by all elements $v \otimes w - w \otimes v$ for $v, w \in V$ . The grading descends: $Sym (V) = ⨁_{k \geq 0} Sym^{k} (V)$ .

The wedge product satisfies $v \land w = - w \land v$ for all $v, w \in V$ (anti-commutativity on degree-1 elements), since $(v + w) \land (v + w) = 0$ expands to $v \land w + w \land v = 0$ . The symmetric product satisfies $v \cdot w = w \cdot v$ (commutativity on degree-1 elements) by construction.

Counterexamples to common slips [Intermediate+]

$Λ^{k} (V)$ is not a subalgebra of $T (V)$ . It is a quotient: the wedge product is not the restriction of the tensor product but the image of the tensor product under the quotient map. In particular, $v \land w \neq = v \otimes w$ ; the former is a coset in $T (V) / I_{ext}$ .
The exterior algebra is finite-dimensional, but the symmetric algebra is not. For $dim V = n$ , the total dimension of $Λ (V)$ is $2^{n}$ (the sum of binomial coefficients). But $dim Sym^{k} (V) = (k n + k - 1) > 0$ for all $k$ , so $Sym (V)$ is infinite-dimensional.
$T (V)$ is non-commutative for $dim V \geq 2$ . In $T (V)$ , the element $v \otimes w - w \otimes v$ is nonzero for linearly independent $v, w$ . Both $Λ (V)$ and $Sym (V)$ address this — one by imposing anti-commutativity ( $v \land w = - w \land v$ ), the other by imposing commutativity ( $v \cdot w = w \cdot v$ ).

Key theorem with proof [Intermediate+]

Theorem (Dimension of exterior powers). Let $V$ be an $n$ -dimensional vector space over a field $K$ . Then for each $0 \leq k \leq n$ : $dim_{K} Λ^{k} (V) = (k n)$ and $Λ^{k} (V) = 0$ for $k > n$ . In particular, $Λ (V)$ has total dimension $2^{n}$ .

Proof. Fix a basis ${e_{1}, \dots, e_{n}}$ for $V$ . For a strictly increasing multi-index $I = (i_{1} < i_{2} < \dots < i_{k})$ with $1 \leq i_{j} \leq n$ , define $e_{I} = e_{i_{1}} \land e_{i_{2}} \land \dots \land e_{i_{k}}$ .

Spanning. Any element of $Λ^{k} (V)$ is a $K$ -linear combination of elements $v_{1} \land \dots \land v_{k}$ with $v_{j} \in V$ . Expanding each $v_{j}$ in the basis and using multilinearity of $\land$ , every such wedge product becomes a linear combination of terms $e_{j_{1}} \land \dots \land e_{j_{k}}$ where the indices need not be increasing or distinct.

For any permutation $σ \in S_{k}$ , the anti-commutativity relation gives $e_{j_{σ (1)}} \land \dots \land e_{j_{σ (k)}} = sgn (σ) \cdot e_{j_{1}} \land \dots \land e_{j_{k}} .$ If two indices among ${j_{1}, \dots, j_{k}}$ are equal, say $j_{a} = j_{b}$ , then swapping the positions $a$ and $b$ introduces a sign of $- 1$ but leaves the element unchanged (same basis vector in both slots), forcing $e_{j_{1}} \land \dots \land e_{j_{k}} = 0$ . Therefore only terms with distinct, strictly increasing indices survive, and the elements ${e_{I} : I strictly increasing, ∣ I ∣ = k}$ span $Λ^{k} (V)$ .

For $k > n$ , any wedge product $e_{j_{1}} \land \dots \land e_{j_{k}}$ involves $k > n$ indices drawn from ${1, \dots, n}$ , so at least two must be equal, giving zero by the argument above.

Linear independence. Suppose $\sum_{I} a_{I} e_{I} = 0$ where the sum runs over all strictly increasing multi-indices of length $k$ . Fix one such multi-index $J = (j_{1} < \dots < j_{k})$ and let $J^{c} = (j_{k + 1} < \dots < j_{n})$ denote the complementary strictly increasing multi-index, so ${j_{1}, \dots, j_{n}}$ is a permutation of ${1, \dots, n}$ .

Set $e_{J^{c}} = e_{j_{k + 1}} \land \dots \land e_{j_{n}} \in Λ^{n - k} (V)$ . Wedging the relation $\sum_{I} a_{I} e_{I} = 0$ with $e_{J^{c}}$ : $I \sum a_{I} e_{I} \land e_{J^{c}} = 0.$

For any $I \neq = J$ , the set $I \cup J^{c}$ contains a repeated index: since $I$ differs from $J$ , there exists an element of $I$ not in $J$ , and this element must lie in $J^{c}$ (the complement of $J$ ). This repeated index forces $e_{I} \land e_{J^{c}} = 0$ .

For $I = J$ , the product $e_{J} \land e_{J^{c}}$ involves every basis vector $e_{1}, \dots, e_{n}$ exactly once, so $e_{J} \land e_{J^{c}} = \pm e_{1} \land \dots \land e_{n}$ , a nonzero element of the one-dimensional space $Λ^{n} (V)$ . Therefore: $a_{J} \cdot (\pm e_{1} \land \dots \land e_{n}) = 0.$ Since $e_{1} \land \dots \land e_{n} \neq = 0$ , we conclude $a_{J} = 0$ . As $J$ was arbitrary, all coefficients vanish. The number of strictly increasing multi-indices of length $k$ from ${1, \dots, n}$ is $(k n)$ . $□$

Bridge. The dimension formula builds toward the Plucker embedding of the Grassmannian, where each $k$ -dimensional subspace of $V$ corresponds to a point in $P (Λ^{k} (V))$ , and appears again in 01.02.10 (tensor product) through the identification of $Λ^{k} (V)$ as a quotient of $V^{\otimes k}$ . The foundational reason is that anti-symmetry reduces the $n^{k}$ tensor products at level $k$ to exactly $(k n)$ independent wedge products. This is exactly the combinatorial content that identifies the exterior algebra with the algebra of alternating multilinear forms, and the bridge is between the algebraic construction (quotient of $T (V)$ ) and the geometric content (oriented $k$ -dimensional volumes). Putting these together with the determinant identification $A v_{1} \land \dots \land A v_{n} = det (A) \cdot v_{1} \land \dots \land v_{n}$ , the exterior algebra becomes the natural home for determinant theory.

Exercises [Intermediate+]

Exercise 4 (medium, symbolic).

Let $dim V = n$ . Prove that $dim Sym^{k} (V) = (k n + k - 1)$ .

Hint

A basis for $Sym^{k} (V)$ is given by monomials $e_{i_{1}} \dots e_{i_{k}}$ with $1 \leq i_{1} \leq i_{2} \leq \dots \leq i_{k} \leq n$ . Count these using the stars-and-bars method: the number of non-decreasing sequences of length $k$ from ${1, \dots, n}$ equals the number of ways to write $k$ as an ordered sum of $n$ non-negative integers.

Answer

A basis for $Sym^{k} (V)$ is the set of monomials ${e_{i_{1}} e_{i_{2}} \dots e_{i_{k}} : 1 \leq i_{1} \leq i_{2} \leq \dots \leq i_{k} \leq n}$ , since commutativity allows reordering of factors.

To count them, associate to each non-decreasing sequence $(i_{1} \leq \dots \leq i_{k})$ the multiplicities $m_{j} = ∣ {l : i_{l} = j} ∣$ for $j = 1, \dots, n$ . Then $m_{j} \geq 0$ and $m_{1} + m_{2} + \dots + m_{n} = k$ . The number of $n$ -tuples $(m_{1}, \dots, m_{n})$ of non-negative integers summing to $k$ is $(k n + k - 1)$ by the stars-and-bars formula. Therefore $dim Sym^{k} (V) = (k n + k - 1)$ .

Exercise 7 (hard, symbolic).

State and prove the universal property of the exterior algebra: for any associative $K$ -algebra $A$ and any $K$ -linear map $f : V \to A$ satisfying $f (v)^{2} = 0$ for all $v \in V$ , there exists a unique $K$ -algebra homomorphism $φ : Λ (V) \to A$ extending $f$ .

Hint

Use the universal property of $T (V)$ : any linear map from $V$ into an algebra extends uniquely to an algebra homomorphism from $T (V)$ . Show that the condition $f (v)^{2} = 0$ forces the kernel to contain $I_{ext}$ .

Answer

By the universal property of $T (V)$ , any $K$ -linear map $f : V \to A$ extends to a unique $K$ -algebra homomorphism $\tilde{φ} : T (V) \to A$ defined by $\tilde{φ} (v_{1} \otimes \dots \otimes v_{k}) = f (v_{1}) \dots f (v_{k})$ . For any $v \in V$ : $\tilde{φ} (v \otimes v) = f (v) \cdot f (v) = f (v)^{2} = 0.$ Therefore $\tilde{φ}$ vanishes on every generator $v \otimes v$ of $I_{ext}$ , and since $\tilde{φ}$ is an algebra homomorphism, it vanishes on the entire ideal $I_{ext}$ . So $\tilde{φ}$ descends to a well-defined $K$ -algebra homomorphism $φ : T (V) / I_{ext} = Λ (V) \to A$ with $φ (v_{1} \land \dots \land v_{k}) = f (v_{1}) \dots f (v_{k})$ .

Uniqueness follows because $V$ generates $Λ (V)$ as a $K$ -algebra (every element is a sum of products of elements from $V$ ), so any algebra homomorphism from $Λ (V)$ is determined by its restriction to $V$ .

Exercise 8 (hard, symbolic).

Prove that $Λ (V)$ is graded-commutative: for homogeneous elements $α \in Λ^{k} (V)$ and $β \in Λ^{l} (V)$ , $α \land β = (- 1)^{k l} β \land α .$

Hint

First verify the formula for degree-1 elements. For general homogeneous elements, reduce to decomposable elements $α = v_{1} \land \dots \land v_{k}$ and $β = w_{1} \land \dots \land w_{l}$ , and count how many sign changes arise from moving $l$ factors past $k$ factors.

Answer

The base case $k = l = 1$ is the anti-commutativity $v \land w = - w \land v = (- 1)^{1} w \land v$ .

For the general case, it suffices to verify on decomposable elements $α = v_{1} \land \dots \land v_{k}$ and $β = w_{1} \land \dots \land w_{l}$ , since these span $Λ^{k} (V)$ and $Λ^{l} (V)$ respectively.

We have $α \land β = v_{1} \land \dots \land v_{k} \land w_{1} \land \dots \land w_{l}$ . To convert this to $β \land α = w_{1} \land \dots \land w_{l} \land v_{1} \land \dots \land v_{k}$ , we must move each of the $l$ vectors $w_{1}, \dots, w_{l}$ past each of the $k$ vectors $v_{1}, \dots, v_{k}$ .

Moving $w_{1}$ from position $k + 1$ to position $1$ requires $k$ transpositions with its left neighbour, each contributing a factor of $- 1$ , giving $(- 1)^{k}$ . Repeating for each of the $l$ vectors contributes a total sign of $((- 1)^{k})^{l} = (- 1)^{k l}$ .

Formally, proceed by induction on $l$ . For $l = 1$ , write $β = w$ . Applying $v_{i} \land w = - w \land v_{i}$ repeatedly: $v_{1} \land \dots \land v_{k} \land w = (- 1)^{k} w \land v_{1} \land \dots \land v_{k} .$ For $l > 1$ , write $β = w_{1} \land β^{'}$ where $β^{'} = w_{2} \land \dots \land w_{l} \in Λ^{l - 1} (V)$ . Then: $α \land (w_{1} \land β^{'}) = (- 1)^{k} w_{1} \land (α \land β^{'}) = (- 1)^{k} (- 1)^{k (l - 1)} w_{1} \land (β^{'} \land α) = (- 1)^{k l} (β \land α)$ by the inductive hypothesis applied to $α$ and $β^{'}$ .

Advanced results [Master]

Theorem 1 (Universal property of $T (V)$ ). The tensor algebra $T (V)$ is the free associative $K$ -algebra on $V$ : for any associative $K$ -algebra $A$ and any $K$ -linear map $f : V \to A$ , there exists a unique $K$ -algebra homomorphism $φ : T (V) \to A$ extending $f$ . The homomorphism is given by $φ (v_{1} \otimes \dots \otimes v_{k}) = f (v_{1}) \dots f (v_{k})$ .

Every algebra generated by $V$ is a quotient of $T (V)$ by a two-sided ideal ^{[Bourbaki Algebre Ch. III]}. The exterior algebra (ideal generated by $v \otimes v$ ) and the symmetric algebra (ideal generated by $v \otimes w - w \otimes v$ ) are the two most fundamental quotients.

Theorem 2 (Determinant via exterior algebra). Let $A : V \to V$ be a linear endomorphism of an $n$ -dimensional vector space. Then $A$ induces $Λ^{k} (A) : Λ^{k} (V) \to Λ^{k} (V)$ for each $k$ , and on the top exterior power: $Λ^{n} (A) (v_{1} \land \dots \land v_{n}) = A v_{1} \land \dots \land A v_{n} = det (A) \cdot v_{1} \land \dots \land v_{n} .$

The determinant is the unique scalar by which $A$ acts on the one-dimensional space $Λ^{n} (V)$ ^{[Cartan 1922]}. This characterisation makes the multiplicative property $det (A B) = det (A) det (B)$ immediate: $Λ^{n} (A B) = Λ^{n} (A) \circ Λ^{n} (B)$ , and composition of scalar multiplications multiplies the scalars.

Theorem 3 (Grassmannian and the Plucker embedding). For a vector space $V$ of dimension $n$ , the Grassmannian $Gr (k, V)$ of $k$ -dimensional subspaces embeds into $P (Λ^{k} (V))$ via the map $W \mapsto [w_{1} \land \dots \land w_{k}]$ , where ${w_{1}, \dots, w_{k}}$ is any basis for $W$ . The image is cut out by the Plucker relations: quadratic equations in the homogeneous coordinates on $P (Λ^{k} (V))$ .

The Plucker embedding realises $Gr (k, V)$ as a projective variety of dimension $k (n - k)$ inside a projective space of dimension $(k n) - 1$ ^{[Griffiths-Harris 1978]}.

Theorem 4 (Hilbert series of $Sym (V)$ ). If $V$ has dimension $n$ over $K$ , the symmetric algebra $Sym (V)$ is isomorphic to the polynomial ring $K [x_{1}, \dots, x_{n}]$ . The Hilbert series with the standard grading is: $H_{Sym (V)} (t) = k \geq 0 \sum dim Sym^{k} (V) t^{k} = \frac{1}{( 1 - t ) ^{n}} .$

The Hilbert series encodes the growth of dimensions across all graded pieces and is the generating function for $(k n + k - 1)$ ^{[Eisenbud 1995]}.

Theorem 5 (Koszul complex). Let $V$ be a vector space of dimension $n$ and let $θ \in V^{*}$ be a linear functional. The Koszul complex $K (θ)$ is the chain complex $0 \to Λ^{n} (V) d_{θ} Λ^{n - 1} (V) d_{θ} \dots d_{θ} Λ^{1} (V) d_{θ} Λ^{0} (V) \to 0$ where $d_{θ} (w_{1} \land \dots \land w_{k}) = \sum_{i = 1}^{k} (- 1)^{i - 1} θ (w_{i}) w_{1} \land \dots \land w_{i} \land \dots \land w_{k}$ . If $θ \neq = 0$ , the Koszul complex is exact.

The Koszul complex is the archetypal linear free resolution and the building block for regular sequences in commutative algebra ^{[Eisenbud 1995]}.

Theorem 6 (Exterior derivative). Let $M$ be a smooth manifold. The exterior derivative $d : Ω^{k} (M) \to Ω^{k + 1} (M)$ is the unique $R$ -linear map on differential forms satisfying: (i) $d (f) = df$ for functions $f \in Ω^{0} (M)$ , (ii) $d (α \land β) = d α \land β + (- 1)^{d e g α} α \land d β$ (graded Leibniz rule), (iii) $d \circ d = 0$ . The de Rham cohomology $H_{dR}^{*} (M) = ker d /, im d$ is the cohomology of the resulting complex.

The exterior derivative generalises grad, curl, and div from vector calculus to a single operator on the exterior algebra of differential forms ^{[Cartan 1922]}.

Theorem 7 (Symmetrisation and PBW). The symmetrisation map $σ : Sym (V) \to T (V)$ defined on degree- $k$ elements by $σ (v_{1} \dots v_{k}) = \frac{1}{k !} τ \in S_{k} \sum v_{τ (1)} \otimes \dots \otimes v_{τ (k)}$ is an injective linear map identifying $Sym (V)$ with the subspace of symmetric tensors in $T (V)$ . As coalgebras, $σ$ is an isomorphism $Sym (V) ≅$ symmetric tensors in $T (V)$ .

This is the special case of the Poincare-Birkhoff-Witt theorem when the Lie algebra is abelian ^{[Bourbaki Algebre Ch. III]}.

Synthesis. The universal property of $T (V)$ is the foundational reason that every algebra generated by $V$ factors through the tensor algebra, and the exterior and symmetric algebras are the two fundamental quotients obtained by imposing anti-commutativity and commutativity respectively. The central insight is that the dimension formula $dim Λ^{k} (V) = (k n)$ and the Hilbert series $(1 - t)^{- n}$ of $Sym (V)$ are the two generating functions governing the growth of these algebras.

Putting these together with the Plucker embedding, which realises the Grassmannian as a subvariety of $P (Λ^{k} (V))$ , and the Koszul complex, which uses the exterior algebra to build the archetypal free resolution, the bridge is between the algebraic construction (graded quotient of $T (V)$ ) and the geometric and topological structures these algebras parametrise. This is exactly the structure that generalises from vector spaces to vector bundles, where $Λ^{k} (E)$ and $Sym^{k} (E)$ are the $k$ -th exterior and symmetric power bundles, and the pattern recurs throughout algebraic topology and differential geometry 01.02.10.

Full proof set [Master]

Proposition 1 (Determinant via wedge product). Let $A : V \to V$ be a linear map on an $n$ -dimensional vector space. Then for any $v_{1}, \dots, v_{n} \in V$ : $A v_{1} \land \dots \land A v_{n} = det (A) \cdot v_{1} \land \dots \land v_{n} .$

Proof. Both sides are alternating multilinear functions of $(v_{1}, \dots, v_{n})$ . Since $Λ^{n} (V)$ is one-dimensional with basis $e_{1} \land \dots \land e_{n}$ , the map $(v_{1}, \dots, v_{n}) \mapsto A v_{1} \land \dots \land A v_{n}$ is a scalar multiple of $(v_{1}, \dots, v_{n}) \mapsto v_{1} \land \dots \land v_{n}$ . Define $det (A)$ to be this scalar.

To compute it, choose a basis ${e_{1}, \dots, e_{n}}$ and write $A e_{j} = \sum_{i} a_{ij} e_{i}$ . Then: $A e_{1} \land \dots \land A e_{n} = (i_{1} \sum a_{i_{1}, 1} e_{i_{1}}) \land \dots \land (i_{n} \sum a_{i_{n}, n} e_{i_{n}}) = i_{1}, \dots, i_{n} \sum a_{i_{1}, 1} \dots a_{i_{n}, n} e_{i_{1}} \land \dots \land e_{i_{n}} .$

Terms with repeated indices vanish. The remaining terms are permutations: $= σ \in S_{n} \sum a_{σ (1), 1} \dots a_{σ (n), n} e_{σ (1)} \land \dots \land e_{σ (n)} = σ \in S_{n} \sum sgn (σ) a_{σ (1), 1} \dots a_{σ (n), n} \cdot e_{1} \land \dots \land e_{n} .$

The coefficient is the Leibniz formula for $det (A)$ . $□$

Proposition 2 (Hilbert series of $Sym (V)$ ). If $dim V = n$ , then $Sym (V) ≅ K [x_{1}, \dots, x_{n}]$ as graded $K$ -algebras, and the Hilbert series is $H (t) = (1 - t)^{- n}$ .

Proof. Choose a basis ${e_{1}, \dots, e_{n}}$ for $V$ . The map $φ : Sym (V) \to K [x_{1}, \dots, x_{n}]$ sending $e_{i} \mapsto x_{i}$ is a surjective $K$ -algebra homomorphism by construction. The monomials ${x_{1}^{a_{1}} \dots x_{n}^{a_{n}} : a_{1} + \dots + a_{n} = k}$ form a basis for the degree- $k$ part of $K [x_{1}, \dots, x_{n}]$ , and their number is $(k n + k - 1)$ , which equals $dim Sym^{k} (V)$ by Exercise 4. Since $φ$ is surjective and the domain and codomain have the same dimension at each graded level, $φ$ is an isomorphism.

The generating function is: $k \geq 0 \sum (k n + k - 1) t^{k} = \frac{1}{( 1 - t ) ^{n}}$ by the generalised binomial theorem: $(1 - t)^{- n} = \sum_{k \geq 0} (k - n) (- t)^{k} = \sum_{k \geq 0} (k n + k - 1) t^{k}$ . $□$

Connections [Master]

Tensor product of modules 01.02.10. The tensor algebra $T (V)$ is built from iterated tensor products $V^{\otimes k}$ , and the universal property of $T (V)$ extends the tensor product's universal property from bilinear maps to multilinear maps of arbitrary arity. The exterior algebra $Λ (V) = T (V) / ⟨ v \otimes v ⟩$ and the symmetric algebra $Sym (V) = T (V) / ⟨ v \otimes w - w \otimes v ⟩$ are both quotients of $T (V)$ by ideals generated by elements in the tensor product. The foundational reason these quotient constructions work is that the tensor product provides the universal bilinear pairing 01.02.10, and iterating it produces all multilinear products.
Vector space 01.01.03. A vector space $V$ over a field $K$ is the raw material from which $T (V)$ , $Λ (V)$ , and $Sym (V)$ are constructed. The dimension of $V$ controls the dimensions of all three algebras: $dim Λ^{k} (V) = (k n)$ , $dim Sym^{k} (V) = (k n + k - 1)$ , and $dim V^{\otimes k} = n^{k}$ . A choice of basis for $V$ determines explicit bases for all three algebras.
Bilinear and quadratic forms 01.01.15. The exterior algebra provides the framework for alternating bilinear forms: a bilinear form $ω$ on $V$ is alternating precisely when it factors through $Λ^{2} (V)$ , meaning $ω (v, w) = \tilde{ω} (v \land w)$ for a linear functional $\tilde{ω}$ on $Λ^{2} (V)$ . A non-degenerate alternating form (symplectic form) on a $2 n$ -dimensional space identifies $V$ with its dual via the wedge product, and the classification of alternating forms parallels the construction of $Λ (V)$ .

Historical & philosophical context [Master]

Grassmann 1844 introduced the exterior product in Die lineale Ausdehnungslehre ^{[Grassmann 1844]}, developing a calculus of extensive quantities that included anti-commuting products, the wedge operation, and the foundations of what is now called the exterior algebra. The work was largely ignored for decades. Clifford 1878 connected Grassmann's exterior algebra with Hamilton's quaternions by introducing a single generating relation $v \otimes v = Q (v)$ for a quadratic form $Q$ , producing what are now called Clifford algebras ^{[Clifford 1878]}. Cartan 1922 systematised the exterior derivative and the calculus of differential forms in Lecons sur les invariants integraux ^{[Cartan 1922]}, establishing the exterior algebra as the natural language for differential geometry and de Rham cohomology.

The modern synthesis, in which $T (V)$ , $Λ (V)$ , and $Sym (V)$ are presented uniformly as quotient algebras of the tensor algebra, crystallised in Bourbaki's Algebre chapters II-III. The identification of the exterior algebra with alternating multilinear forms and the symmetric algebra with polynomial rings made these constructions central to both commutative algebra and algebraic geometry.

Bibliography [Master]

@book{Grassmann1844,
  author = {Grassmann, Hermann G.},
  title = {Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik},
  publisher = {Wigand},
  year = {1844},
  address = {Leipzig},
}

@article{Clifford1878,
  author = {Clifford, William K.},
  title = {Applications of Grassmann's extensive algebra},
  journal = {American Journal of Mathematics},
  volume = {1},
  year = {1878},
  pages = {350--358},
}

@book{Cartan1922,
  author = {Cartan, Elie},
  title = {Lecons sur les invariants integraux},
  publisher = {Hermann},
  year = {1922},
  address = {Paris},
}

@book{Bourbaki1989,
  author = {Bourbaki, Nicolas},
  title = {Algebra I: Chapters 1--3},
  publisher = {Springer},
  year = {1989},
}

@book{Greub1978,
  author = {Greub, Werner H.},
  title = {Multilinear Algebra},
  edition = {2nd},
  publisher = {Springer},
  year = {1978},
}

@book{Eisenbud1995,
  author = {Eisenbud, David},
  title = {Commutative Algebra with a View Toward Algebraic Geometry},
  publisher = {Springer},
  year = {1995},
  series = {Graduate Texts in Mathematics 150},
}

@book{GriffithsHarris1978,
  author = {Griffiths, Phillip and Harris, Joseph},
  title = {Principles of Algebraic Geometry},
  publisher = {Wiley},
  year = {1978},
}

Prerequisites

01.02.10
01.01.03

Tier anchors

beginner: Greub Multilinear Algebra informal; building products from a vector space
intermediate: Greub 1978 Multilinear Algebra; Bourbaki Algebre Ch. III
master: Grassmann 1844 Ausdehnungslehre; Cartan 1922 Lecons; Bourbaki Algebre Ch. III

References

TODO_REF
Grassmann — Die lineale Ausdehnungslehre · 1844, Leipzig
TODO_REF
Clifford — Applications of Grassmann's extensive algebra · Amer. J. Math. 1, 1878
TODO_REF
Cartan — Lecons sur les invariants integraux · 1922, Hermann
TODO_REF
Greub — Multilinear Algebra · Springer, 1978
TODO_REF
Bourbaki — Algebre Ch. III · Springer
TODO_REF
Eisenbud — Commutative Algebra with a View Toward Algebraic Geometry · Ch. 2
TODO_REF
Griffiths-Harris — Principles of Algebraic Geometry · Wiley, 1978

Reviewer

TBD

Estimated time

beginner: 15m
intermediate: 40m
master: 75m