Invariant Determinants and Differential Forms IITG PHYSICS SEMINAR

Srikanth K.V.

Department of Mathematics, IIT Guwahati

Invariant Determinants and Differential Forms – p. 1 Overview of the talk

1. Laplace’s formula and Leibnitz’s formula 2. Properties of determinants 3. Determinant of a linear operator 4. A dilemmatic example 5. Volume in Euclidean spaces 6. Abstract volume functions in vector spaces 7. Geometric deﬁnition of determinant function 8. Advantages of geometric deﬁnition 9. Differential forms

Invariant Determinants and Differential Forms – p. 2 Laplace's and Leibnitz's formula

Let A be an n × n matrix over a ﬁeld F . Then

n det A = a C X ij ij i=1 n = (−1)i+ja M X ij ij i=1

where Cij is cofactor of aij and Mij is the (ij)-th minor of A.

Invariant Determinants and Differential Forms – p. 3 Laplace's and Leibnitz's formula

Let A be an n × n matrix over a ﬁeld F . Then

n det A = sign(σ) a X Y iσ(i) σ∈Sn i=1 This formula probably has its roots in Cramer’s Rule type solutions to a linear system Ax = b. Remark: This formula makes sense even if F is a commutative ring.

Invariant Determinants and Differential Forms – p. 4 Properties of determinants

1. det(AB) = det A det B 2. If any row/column of A is a linear combination of the remaining rows/columns of A then det A = 0. 3. If B is a matrix obtained by interchanging any two rows or columns of matrix A then det A = − det B 4. det(kA) = kn det(A), for any k ∈ F 5. det A = det AT 6. det A−1 = (det A)−1

Invariant Determinants and Differential Forms – p. 5 Properties of determinants

Remarks: The proofs of properties 1, 2 and 3 are usually via the deﬁning formulae. They are somewhat hard and unilluminating. The invariant or geometric deﬁnition we are going to see makes the proofs of these a triviality.

Invariant Determinants and Differential Forms – p. 6 Determinant of a linear operator

We know that determinants of similar matrices are equal.

Invariant Determinants and Differential Forms – p. 7 Determinant of a linear operator

We know that determinants of similar matrices are equal. If f is a linear operator on a vector space V the following is well–deﬁned det f = det (matrix of f with respect to any basis B of V)

Invariant Determinants and Differential Forms – p. 7 Determinant of a linear operator

We know that determinants of similar matrices are equal. If f is a linear operator on a vector space V the following is well–deﬁned det f = det (matrix of f with respect to any basis B of V) The above deﬁnition does not tell us which property of f is captured by det f

Invariant Determinants and Differential Forms – p. 7 A dilemmatic example

Invariant Determinants and Differential Forms – p. 8 Volume in euclidean spaces

Invariant Determinants and Differential Forms – p. 9 Abst. vol. functions in vector spaces

Invariant Determinants and Differential Forms – p. 10 Abst. vol. functions in vector spaces

Let V be a vector space over a field F . Define a k-volume function as a function µ : V k → F which satisfies 1. µ is k-linear 2. µ is alternating i.e. µ(a1, a2, ..., ak) = 0, whenever ai = aj for i =6 j, 1 ≤ i, j ≤ k ⋆ µ is alternating ⇒ µ is skew symmetric. k ⋆ If {ai}i=1 is a linearly dependent set then µ(a1, ..., ak) = 0

Invariant Determinants and Differential Forms – p. 10 Abst. vol. functions in vector spaces

Note that if σ ∈ Sk, the symmetric group over k symbols, then

µ(aσ(1), aσ(2),...,aσ(k)) = sign(σ) µ(a1, a2,...,ak)

Invariant Determinants and Differential Forms – p. 11 Abst. vol. functions in vector spaces

Note that if σ ∈ Sk, the symmetric group over k symbols, then

µ(aσ(1), aσ(2),...,aσ(k)) = sign(σ) µ(a1, a2,...,ak) Let dim V = n. For any k ∈ N, deﬁne Λk(V ) = {µ : V k → F : µ is k−linear and alternating}

Invariant Determinants and Differential Forms – p. 11 Abst. vol. functions in vector spaces

Note that if σ ∈ Sk, the symmetric group over k symbols, then

µ(aσ(1), aσ(2),...,aσ(k)) = sign(σ) µ(a1, a2,...,ak) Let dim V = n. For any k ∈ N, deﬁne Λk(V ) = {µ : V k → F : µ is k−linear and alternating} It can be veriﬁed that Λk(V ) is a vector space over F and

n k  k if k ≤ n dim Λ (V ) =  0 if k>n  Invariant Determinants and Differential Forms – p. 11 Determinant using volume functions

Since Λn(V ) is one dimensional, any two (non–zero) volume functions on V are multiples of each other.

Invariant Determinants and Differential Forms – p. 12 Determinant using volume functions

Since Λn(V ) is one dimensional, any two (non–zero) volume functions on V are multiples of each other. Let f : V → V be a linear transformation and n n {bi}i=1 a basis of V . Pick any 0 =6 µ ∈ Λ (V ).

Invariant Determinants and Differential Forms – p. 12 Determinant using volume functions

µ(f(b1),...,f(bn))

µ(b1,...,bn) is independent of both the choice of basis and the choice of µ.

Invariant Determinants and Differential Forms – p. 12 Geometric defn. of the det. function

So deﬁne det f as µ(f(b ), ..., f(b )) det f = 1 n µ(b1, ..., bn)

n using any basis {bi}i=1 of V and any nonzero µ ∈ Λn(V )

Invariant Determinants and Differential Forms – p. 13 Geometric defn. of the det. function

So deﬁne det f as µ(f(b ), ..., f(b )) det f = 1 n µ(b1, ..., bn)

n using any basis {bi}i=1 of V and any nonzero µ ∈ Λn(V ) Remark: The above deﬁnition is faithful to the interpretation of determinant of a linear transformation as "volume expansion factor" Note that if h is a singular transformation, then det h = 0. Invariant Determinants and Differential Forms – p. 13 Consistency with known defn.

Given n vectors b1,...,bn in V , write them as a column B. Express the entries in terms of the standard basis E via B = CE. The usual determinant of the matrix C can be viewed as the value of µ(b1,...,bn) for an element µ in Λn(V ). Using this one can prove consistency.

Invariant Determinants and Differential Forms – p. 14 Determinant of a composition

If f or g is singular then g ◦ f is singular and det f = 0 or det g = 0 and det g ◦ f = 0. So det(g ◦ f) = det g det f.

Invariant Determinants and Differential Forms – p. 15 Determinant of a composition

If f or g is singular then g ◦ f is singular and det f = 0 or det g = 0 and det g ◦ f = 0. So det(g ◦ f) = det g det f. If f and g are nonsingular then

µ(g ◦ f(b ), ..., g ◦ f(b ) det(g ◦ f)= 1 n µ(b1, ..., bn) µ(g(f(b )), ..., g(f(b )) µ(f(b ), ..., f(b ) = 1 n 1 n µ(f(b1), ..., f(bn)) µ(b1, ..., bn) = det g det f

Invariant Determinants and Differential Forms – p. 15 Search for Integrands

Question: If N is a k-dimensional submanifold of a differential manifold M, what is the right object which we can integrate over N?

Invariant Determinants and Differential Forms – p. 16 Search for Integrands

Question: If N is a k-dimensional submanifold of a differential manifold M, what is the right object which we can integrate over N? Hint: We have encountered a similar situation in vector calculus when we studied line integrals (like work) and surface integrals (like ﬂux).

Invariant Determinants and Differential Forms – p. 16 Search for Integrands

Invariant Determinants and Differential Forms – p. 16 Differential forms

Observation: In line integrals, integrand :: linear function & In surface integrals, integrand :: bilinear alternating function (on tangent spaces)

Invariant Determinants and Differential Forms – p. 17 Differential forms

Observation: In line integrals, integrand :: linear function & In surface integrals, integrand :: bilinear alternating function (on tangent spaces) Pick a smoothly varying ω on M such that for k each p ∈ M, ωp ∈ Λ (TpM) where k is the dimension of the submanifold N.

Invariant Determinants and Differential Forms – p. 17 Differential forms

Invariant Determinants and Differential Forms – p. 17 Generalizations

Grad, Curl and Divergence :: Exterior derivative Theorems of Green, Gauss and Stokes :: General Stokes Theorem Path independence and potential functions :: Poincare Lemma Failure of Poincare Lemma :: Cohomology groups Relation between Cohomology groups and Shape of M :: Theorem of de Rham

Invariant Determinants and Differential Forms – p. 18