Representations and Line Bundles

We have seen that the splitting principle allows us to regard the Chern classes of a vector bundle E as (symmetric) polynomials in the first Chern classes of the line bundles of a flag bundle associated to E. Given an m-dimensional vector space V, the spaceF `(V) of all complete flags on V can be identified with GL(V)/P, whereP is the group of upper triangular matrices. (This follows because GL(V) is transitive on the flags and P, the stabilizer of the standard flag 0 ⊂ he1i ⊂ he1, e2i ⊂ · · · ⊂ he<sub>1</sub>, . . . , e<sub>m</sub>i=V, is isomorphic to the stabilizer of any given flag.) We can associate any one-dimensional representation χ : P → C<sup>∗</sup> to an equivariant line bundle over the flag manifold F `(V) as follows:

L(χ) = GL(V)×C/((gp, z)∼(g, χ(p)z)) (8.14) for g ∈ GL(V), p ∈ P, and z ∈ C. The projection of L(χ) onto F `(V) is just (g, z)→^π (gP). Under the action of GL(V) given by h(gp, z) = (hgp, z), the following diagram commutes:

L(χ) −−−−→^GL(V) L(χ)

π



 y



 y^π F `(V) −−−−→<sup>GL(V)</sup> F `(V),

since (hgp, z) = (hg, χ(p)z). The line bundleL(χ) is thus equivariant with respect to the bundle projection.

Conversely, suppose L is an equivariant line bundle over F `(V). Then P acts on the fiber over eP, so this fiber is a one-dimensional representation χ of P. Let us show that the line bundle L(χ) corresponding to this representation is isomorphic to L. Lety∈Llie in the fiber overeP. Then we claim that the mapr :L(χ)→Lgiven by r(g, z) = z(g·y) is an isomorphism. We have r(gp, z) = z(gp·y) = z(gχ(p)y) = r(g, χ(p)z), so this map is well-defined. Since G acts transitively on the fibers of L, and multiplication by z is a surjective map on any given fiber, r is surjective. For injectivity, suppose that z1(g1 ·y) = z2(g2 ·y). If z1 6= 0, then y = z⁻₁¹z2g₁⁻¹g2 ·y, so g⁻¹₁ g₂ ∈ P. This means that g₂ = g₁p for some p ∈ P and z⁻¹₁ z₂χ(p) = 1, so z2 = z1χ(p)⁻¹. Thus, as elements of L(χ), (g2, z2) = (g1p, χ(p)⁻¹z1) = (g1, z1), so r is indeed injective. The correspondence between line bundles and one-dimensional representations of P is therefore a bijection.

We can identify the characters χwith line bundles on a flag manifoldF `(V) more explicitly. Consider the tautological filtration[41]

0 =U0 ⊂U1 ⊂U2 ⊂ · · · ⊂Um =F `(V)×V (8.15) of vector bundles overF `(V), whereF `(V)×V is the product bundle, and Uk is the k-dimensional bundle over F `(V) whose fiber over a flag V₁ ⊂ · · · ⊂ V_m is V_k. It follows from the splitting principle that the cohomology ringH^∗(F `(V)) is generated by the first Chern classes of the line bundles L_i =U_i/U_i−1, setting x_i =c₁(L₁). The identity matrix fixes the standard flag {e1, . . . , em}. Therefore, over eP, the fiber of L_i is V_i/V_i−1, where V_i = he₁, . . . , e_ii. If v = Pi

k=1α_ie_i ∈ V_i and p ∈ P, then p·v =w+piiei, wherew∈Vi−1 andpiiis the ith diagonal entry ofp. We have shown

the following.

Theorem 8.4.1 If L_i is the line bundle over a flag manifold defined as above, then the character χ associated to L_i is the map taking p to p_ii.

Let us adapt this machinery to the problem at hand. Recall that we have two complex vector spaces Aand B of dimensionsd_Aand d_B, respectively, together with a map φ: Grk(A)→Gr_kdB(A⊗B) given byφ(V) =V ⊗B. We wish to compute the action of the induced map φ^∗ :H^∗(Gr_kdB(A⊗B))→H^∗(Gr_k(A)).

LetQ_AandQ_ABbe the quotient bundles of Theorem 8.2.2 over the Grassmannians Gr_k(A) and Gr_kdB(A⊗B) respectively. The Chern classes of these bundles are the classes of the special Schubert varieties in the cohomology rings. By the splitting principle, the associated flag bundles Fl(A) and Fl(A ⊗ B) have pullbacks which split as a direct sum of line bundles L_i of the respective tautological filtrations. The cohomology of the Grassmannians embeds in the cohomology of these pullbacks, so we may determine φ^∗ by its action on the Chern classes of the pullback bundle of Fl(A⊗B).

It follows from the definition of pullback bundles that the bundle φ^∗(L(χ_i)) is the set of triples (gP_A, φ(g), z) ∈ GL(A)/P_A ×GL(A⊗B)×C with the identification (gP_A, φ(g ·p), z) ∼ (gP_A, φ(g), χ(φ(p))z). This means that φ^∗(L(χ_i)) = L(φ^∗(χ_i)).

And the pullback of the map induced by φ on the characters of the group P_AB is readily computed: for a matrix X ∈ P_A, and the character χ_i taking a matrix to its ith diagonal entry, we have φ^∗(χ_i)(X) = χ_i(φ(X)) = χ_i(X ⊗I) = χ_di/dBe(X). So φ^∗(χ_i) = χ_di/dBe. Now we can calculate the action ofφ^∗ on the Chern classes:

Theorem 8.4.2 φ^∗(x_i) = c₁(L(χ_di/dBe)).

Proof

φ^∗(x_i) = φ^∗(c₁(L(χ_i))) (8.16)

= c₁(φ^∗(L(χ_i))) (8.17)

= c₁(L(φ^∗(χ_i))) (8.18)

= c₁(L(χ_di/dBe)). (8.19)

2

Chapter 9

Determining the Inequalities

In this chapter we use our knowledge of howφ^∗behaves to explicitly derive inequalities relating the spectra ofρ_AB and ofρ_A. We work out some examples in low dimensions.

We also restate how to obtain the inequalities in the language of representation theory.

We discuss recent progress in symplectic geometry that shows that the inequalities derived using our method are sufficient. Finally, we prove that ifd_B ≥ ¹₂d²_A, then the inequalities simplify greatly.

9.1 Putting It All Together

Let ρA = TrBρAB, and let λ, µ, and ˜λ denote the spectra of ρAB, −ρAB, and ρA, respectively. Theorem 6.2.2 can be interpreted cohomologically as saying that if

φ^∗(σ_π)∪σ˜_ν 6= 0, (9.1)

where σ_π ∈ H^∗(Gr(kd_B, d_Ad_B)) and ˜σ_ν ∈ H^∗(Gr(k, d_a)) are Schubert classes, then the spectra µ and ˜λ must satisfy the inequalities

Xν(i)˜λ_i+X

π(i)µ_i ≤0. (9.2)

Now φ^∗(σ_π) is an integer combination of Schubert classes, φ^∗(σ_π) = X

i

n_iσ˜_πi. (9.3)

For each of these classes, ˜σ_πi ∪σ˜_ν 6= 0 iff ν contains the complement of π_i in the k×(n−k) rectangle. But if we consider the case whereν is in fact the complement ofπ_i, then we see that the Inequalities 9.2 are the strongest in this case; for any other ν⁰ ⊃ν, the inequalities determined by ν⁰ are implied by the inequalities determined byν. So it is sufficient to consider complements of each Schubert class ˜σ_πi contained in φ^∗(σ_π), in order to obtain the inequalities relating −ρ_AB and ρ_A. Now if µ is the spectrum of −ρ_AB, then the spectrum λ of ρ_AB is given by λ_i =−µ_dA−i+1 (since the ordering of the eigenvalues is reversed). Given binary strings π,πˆ ∈ ^dA_k^dB

satisfying ˆ

π(i) =π(d_Ad_B−i+ 1), so that ˆπ is simply the string π in reverse, the Schubert cell S_ˆπ corresponds to the complementary partition to that of S_π. This means that we obtain inequalities

Xν(i)˜λ_i ≤X

π(i)λ_i (9.4)

whenever φ^∗(σ_ˆπ) contains σ_ˆν (where ˆν is the complementary partition to ν) as a summand. It then follows that Inequalities 9.4 are obtained wheneverφ^∗(σ_π) contains σ_ν as a summand.

Theorem 8.2.2 says that the lth Chern class c_l(Q) of the universal quotient bun- dle Q over the Grassmannian Gr(k, n) is equal to the special Schubert class σ_l ∈ H^∗(Gr(k, n)). And the splitting principle allows us to conclude that

c_l(Q) =e_l(x₁, . . . , x_n−k), (9.5) where x_i =c₁(L_i) is the first Chern class of theith split component of f^∗(Q), ande_l is the lth elementary symmetric polynomial. Because the special Schubert classesσ_l generate the cohomology ring, we therefore have a surjective ring homomorphism

ψ˜: Λ_n−k →H^∗(Gr(k, n)) e_l(x₁, . . . x_n−k) 7→σ_l.

We may compose the map ˜ψ with the involution ω : Λn−k → Λn−k, ω(ek) = hk, to

obtain a map

ψ : Λ_n−k →H^∗(Gr(k, n)) h_l(x₁, . . . x_n−k) 7→σ_l.

Now, by the Pieri rule, it follows that for any partition λ, ψ(s_λ(x₁, . . . , x_n−k)) =σ_λ. Thus, we may determine how φ^∗ acts on H^∗(Gr(kd_B, d_Ad_B)) by determining how the map x_i 7→x_di/dBe acts on Schur functions.