I'll start with the second question first. Your question is to describe the tangent and cotangent bundles of projective spaces or a Grassmannian, and their exterior powers. The Grassmannian $\text{Gr}(k,n)$ is also the Grassmannian $\text{Gr}(n-k,n)$, so you could say that it has two tautological bundles $A$ and $B$, of dimension $k$ and $n-k$. Their direct sum is an $n$-dimensional trivial bundle. My intuition from the geometry is that the tangent bundle is $\text{Hom}(A,B)$. You can also write this as $A^* \otimes B$.

Now suppose that $k=1$ so that it is $\mathbb{C}P^n$. Then $A = \mathcal{O}(-1)$ (if I have my signs right) and $B$ is the quotient of $(n+1)\mathcal{O}(0)$ by $A$. So $A^* \otimes B$ is a quotient of $(n+1)\mathcal{O}(1)$ by $\mathcal{O}(0)$. This is thus a resolution of the tangent bundle by direct sums of line bundles. The cotangent bundle has a dual resolution, and you can explode a resolution like this to obtain a resolution of the exterior powers.

This realization of the tangent bundle of $\mathbb{C}P^n$ lets you compute its Chern classes. I think that the values of the Chern classes tell you that it isn't more directly a direct sum of line bundles. When $n=2$, Maple tells me that the Chern classes of the line bundles would be irrational.

I imagine that this is discussed better in some more recent textbook, but Google finds an old paper by Hangan with the same statement that the Grassmannian tangent bundle is a tensor product. The paper considers the geometric implications of having some tensor factorization of the tangent bundle of a manifold in general.

I would guess that you can't describe the tangent bundle of a general Grassmannian in terms of line bundles. The line bundles yield a certain subgroup of the semigroup of possible total Chern classes. I would think that this subgroup misses the Chern classes of the relevant bundles by a mile. (But I don't know a rigorous argument or a reference.)

I'll start with the second question first. Your question is to describe the tangent and cotangent bundles of projective spaces or a Grassmannian, and their exterior powers. The Grassmannian $\text{Gr}(k,n)$ is also the Grassmannian $\text{Gr}(n-k,n)$, so you could say that it has two tautological bundles $A$ and $B$, of dimension $k$ and $n-k$. Their direct sum is an $n$-dimensional trivial bundle. My intuition from the geometry is that the tangent bundle is $\text{Hom}(A,B)$. You can also write this as $A^* \otimes B$.

Now suppose that $k=1$ so that it is $\mathbb{C}P^n$. Then $A = \mathcal{O}(-1)$ (if I have my signs right) and $B$ is the quotient of $(n+1)\mathcal{O}(0)$ by $A$. So $A^* \otimes B$ is a quotient of $(n+1)\mathcal{O}(1)$ by $\mathcal{O}(0)$. This is thus a resolution of the tangent bundle by direct sums of line bundles. The cotangent bundle has a dual resolution, and you can explode a resolution like this to obtain a resolution of the exterior powers.

This realization of the tangent bundle of $\mathbb{C}P^n$ lets you compute its Chern classes. I think that the values of the Chern classes tell you that it isn't more directly a direct sum of line bundles. When $n=2$, Maple tells me that the Chern classes of the line bundles would be irrational. (Edit: As David Speyer explains, you can confirm with pen and paper that the Chern class doesn't factor completely, and doesn't even have a linear factor when $n$ is odd.)

I imagine that this is discussed better in some more recent textbook, but Google finds an old paper by Hangan with the same statement that the Grassmannian tangent bundle is a tensor product. The paper considers the geometric implications of having some tensor factorization of the tangent bundle of a manifold in general.

I would guess that you can't describe the tangent bundle of a general Grassmannian in terms of line bundles. The line bundles yield a certain subgroup of the semigroup of possible total Chern classes. I would think that this subgroup misses the Chern classes of the relevant bundles by a mile. (But I don't know a rigorous argument or a reference.)

I'll start with the second question first. Your question is to describe the tangent and cotangent bundles of projective spaces or a Grassmannian, and their exterior powers. The Grassmannian $\text{Gr}(k,n)$ is also the Grassmannian $\text{Gr}(n-k,n)$, so you could say that it has two tautological bundles $A$ and $B$, of dimension $k$ and $n-k$. Their direct sum is an $n$-dimensional trivial bundle. My intuition from the geometry is that the tangent bundle is $\text{Hom}(A,B)$. You can also write this as $A^* \otimes B$.

Now suppose that $k=1$ so that it is $\mathbb{C}P^n$. Then $A = \mathcal{O}(-1)$ (if I have my signs right) and $B$ is the quotient of $(n+1)\mathcal{O}(0)$ by $A$. So $A^* \otimes B$ is a quotient of $(n+1)\mathcal{O}(1)$ by $\mathcal{O}(0)$. This is thus a resolution of the tangent bundle by direct sums of line bundles. The cotangent bundle has a dual resolution, and you can explode a resolution like this to obtain a resolution of the exterior powers.

This realization of the tangent bundle of $\mathbb{C}P^n$ lets you compute its Chern classes. I think that the values of the Chern classes tell you that it isn't more directly a direct sum of line bundles. When $n=2$, Maple tells me that the Chern classes of the line bundles would be irrational.

I'll start with the second question first. Your question is to describe the tangent and cotangent bundles of projective spaces or a Grassmannian, and their exterior powers. The Grassmannian $\text{Gr}(k,n)$ is also the Grassmannian $\text{Gr}(n-k,n)$, so you could say that it has two tautological bundles $A$ and $B$, of dimension $k$ and $n-k$. Their direct sum is an $n$-dimensional trivial bundle. My intuition from the geometry is that the tangent bundle is $\text{Hom}(A,B)$. You can also write this as $A^* \otimes B$.

Now suppose that $k=1$ so that it is $\mathbb{C}P^n$. Then $A = \mathcal{O}(-1)$ (if I have my signs right) and $B$ is the quotient of $(n+1)\mathcal{O}(0)$ by $A$. So $A^* \otimes B$ is a quotient of $(n+1)\mathcal{O}(1)$ by $\mathcal{O}(0)$. This is thus a resolution of the tangent bundle by direct sums of line bundles. The cotangent bundle has a dual resolution, and you can explode a resolution like this to obtain a resolution of the exterior powers.

This realization of the tangent bundle of $\mathbb{C}P^n$ lets you compute its Chern classes. I think that the values of the Chern classes tell you that it isn't more directly a direct sum of line bundles. When $n=2$, Maple tells me that the Chern classes of the line bundles would be irrational.

I imagine that this is discussed better in some more recent textbook, but Google finds an old paper by Hangan with the same statement that the Grassmannian tangent bundle is a tensor product. The paper considers the geometric implications of having some tensor factorization of the tangent bundle of a manifold in general.

I would guess that you can't describe the tangent bundle of a general Grassmannian in terms of line bundles. The line bundles yield a certain subgroup of the semigroup of possible total Chern classes. I would think that this subgroup misses the Chern classes of the relevant bundles by a mile. (But I don't know a rigorous argument or a reference.)