Monodromy and Real Wronskians Jake Levinson (Simon Fraser - - PowerPoint PPT Presentation

Monodromy and Real Wronskians

Jake Levinson (Simon Fraser University) joint with Kevin Purbhoo (U. Waterloo) ICERM September 2, 2020

Parametric curves and Wronskians

◮ Parametric curve φ : P1 → Pk:

t → φ(t) = [ f0(t) : · · · : fk(t) ], where fi(t) ∈ C[t]≤n.

◮ The Wronskian of f0, . . . , fk is given by

Wr(f0, . . . , fk) = det      f0(t) · · · fk(t) f ′

0(t)

· · · f ′

k(t)

. . . ... . . . f (k) (t) · · · f (k)

k

(t)     

Parametric curves and Wronskians

◮ Parametric curve φ : P1 → Pk:

t → φ(t) = [ f0(t) : · · · : fk(t) ], where fi(t) ∈ C[t]≤n.

◮ The Wronskian of f0, . . . , fk is given by

Wr(f0, . . . , fk) = det      f0(t) · · · fk(t) f ′

0(t)

· · · f ′

k(t)

. . . ... . . . f (k) (t) · · · f (k)

k

(t)     

◮ Detects flexes: t such that φ, φ′, φ′′, . . . , φ(k) is linearly

dependent (e.g. inflection point, cusp, ...)

◮ Simple flex: Rank deficiency at φ(k), fixed at φ(k+1).

What is a simple flex in P3?

◮ C meets its tangent line to order 2 (not special) ◮ C meets its tangent plane to order 3 + 1 = 4 (flex!)

The Wronski problem

Wronski problem

Describe the curves φ with a given Wronskian.

The Wronski problem

Wronski problem

Describe the curves φ with a given Wronskian. Basic combinatorial question: how many?

Theorem (Classical)

There are only finitely-many parametric curves φ with flexes at prescribed ti ∈ P1 (up to PGLk+1).

The Wronski problem

Wronski problem

Describe the curves φ with a given Wronskian. Basic combinatorial question: how many?

Theorem (Classical)

There are only finitely-many parametric curves φ with flexes at prescribed ti ∈ P1 (up to PGLk+1). Deep connection to Schubert calculus: The number of such φ (counted with multiplicity) is the number of standard Young tableaux: SYT( ) =

• 1 2 4

3 5 6 , 1 3 5 2 4 6 , · · ·

Over R, things are remarkably nice!

Shapiro–Shapiro Conjecture (’95) / M–T–V Theorem:

Theorem (Mukhin–Tarasov–Varchenko ’05, ’09)

If Wr(φ) has all real roots, then φ itself is defined over R (up to coordinate change on Pk). Very unusual real algebraic geometry problem with real solutions!

Over R, things are remarkably nice!

Shapiro–Shapiro Conjecture (’95) / M–T–V Theorem:

Theorem (Mukhin–Tarasov–Varchenko ’05, ’09)

If Wr(φ) has all real roots, then φ itself is defined over R (up to coordinate change on Pk). Very unusual real algebraic geometry problem with real solutions!

Theorem (L–Purbhoo ’19)

Let Wr(φ) have n1 distinct real roots, n2 complex conjugate pairs. Over R, the number of such φ, counted with signs, is the symmetric group character χ (2n2, 1n1). Recovers M–T–V in the case n2 = 0.

The Wronski map Gr → PN

Consider the Grassmannian of subspaces f0, . . . , fk ⊆ C[t]≤n.

The Wronski map Gr → PN

Consider the Grassmannian of subspaces f0, . . . , fk ⊆ C[t]≤n. Up to scalar, the Wronskian depends only on f0, . . . , fk: Wr(f0, . . . , fk) = det    f0(t) · · · fk(t) . . . ... . . . f (k) (t) · · · f (k)

k

(t)   

The Wronski map Gr → PN

Consider the Grassmannian of subspaces f0, . . . , fk ⊆ C[t]≤n. Up to scalar, the Wronskian depends only on f0, . . . , fk: Wr(f0, . . . , fk) = det    f0(t) · · · fk(t) . . . ... . . . f (k) (t) · · · f (k)

k

(t)    Gives the Wronski map: Wr : Gr(k+1, C[t]≤n) → P(C[t]≤(k+1)(n−k)), f0, . . . , fk → Wr(f0, . . . , fk). Note: Fiber of Wr = set of all φ with specified flexes.

The Wronski map Gr → PN

Consider the Grassmannian of subspaces f0, . . . , fk ⊆ C[t]≤n. Up to scalar, the Wronskian depends only on f0, . . . , fk: Wr(f0, . . . , fk) = det    f0(t) · · · fk(t) . . . ... . . . f (k) (t) · · · f (k)

k

(t)    Gives the Wronski map: Wr : Gr(k+1, C[t]≤n) → P(C[t]≤(k+1)(n−k)), f0, . . . , fk → Wr(f0, . . . , fk). Note: Fiber of Wr = set of all φ with specified flexes. Aside #1: Fibers are intersections of Schubert varieties.

Shapiro–Shapiro / M–T–V, geometric statement

Theorem (Mukhin–Tarasov–Varchenko ’05, ’09)

If Wr(φ) has distinct real roots, then every point of the fiber is real and reduced.

Shapiro–Shapiro / M–T–V, geometric statement

Theorem (Mukhin–Tarasov–Varchenko ’05, ’09)

If Wr(φ) has distinct real roots, then every point of the fiber is real and reduced. No collisions: they would introduce nearby complex solutions.

Shapiro–Shapiro / M–T–V, geometric statement

Theorem (Mukhin–Tarasov–Varchenko ’05, ’09)

If Wr(φ) has distinct real roots, then every point of the fiber is real and reduced. No collisions: they would introduce nearby complex solutions. Many consequences:

◮ The Wronski map is a covering map over the locus of

distinct real roots:

◮ UCN(RP1) := {sets of n distinct points on RP1}

⊆ P(R[t]≤n).

Shapiro–Shapiro / M–T–V, geometric statement

Theorem (Mukhin–Tarasov–Varchenko ’05, ’09)

If Wr(φ) has distinct real roots, then every point of the fiber is real and reduced. No collisions: they would introduce nearby complex solutions. Many consequences:

◮ The Wronski map is a covering map over the locus of

distinct real roots:

◮ UCN(RP1) := {sets of n distinct points on RP1}

⊆ P(R[t]≤n).

◮ Fiber cardinality is exactly #SYT(

)

◮ Each φ is canonically identified by a tableau [Purbhoo ’09].

Configuration spaces of RP1 and CP1

◮ UCN(RP1) is much simpler than UCN(CP1):

◮ Fundamental group π1(UCN(RP1)) ∼

= Z by rotation by 2π

N .

◮ Open subset UCN(R) is a simplex.

Labeling fibers and monodromy

◮ Purbhoo ’09: Over UCN(R), label sheets by tableaux.

◮ Label a “limit fiber” near “{0, 0, . . . , 0}” /

∈ UCN(R).

◮ Orders of vanishing of Pl¨

ucker coordinates

1 3 4 2 5 6

Labeling fibers and monodromy

◮ Purbhoo ’09: Over UCN(R), label sheets by tableaux.

◮ Label a “limit fiber” near “{0, 0, . . . , 0}” /

∈ UCN(R).

◮ Orders of vanishing of Pl¨

ucker coordinates

1 3 4 2 5 6

◮ Monodromy over ∞ acts by combinatorial bijections!

RP1 ∞ “tableau promotion”

Labeling fibers and monodromy

◮ Purbhoo ’09: Over UCN(R), label sheets by tableaux.

◮ Label a “limit fiber” near “{0, 0, . . . , 0}” /

∈ UCN(R).

◮ Orders of vanishing of Pl¨

ucker coordinates

1 3 4 2 5 6

◮ Monodromy over ∞ acts by combinatorial bijections!

RP1 ∞ “tableau promotion” RP1 ∞ “tableau evacuation”

Aside #2:

◮ Parallel story over M0,N(R) (Kamnitzer, Speyer, Rybnikov) ◮ Topology, genus of curves in Gr(k+1, n+1) (L, Gillespie–L) ◮ Orthogonal Grassmannians (Purbhoo, Gillespie–L–Purbhoo)

A challenge and a new approach

Theorem (Mukhin–Tarasov–Varchenko ’05, ’09)

If Wr(φ) has all real roots, then φ is defined over R (up to change

• f coordinates).

Challenge for geometers:

◮ M–T–V proof uses the Bethe ansatz ◮ Subsequent geometry work used M–T–V as black box. ◮ Many open generalizations of interest!

A challenge and a new approach

Theorem (Mukhin–Tarasov–Varchenko ’05, ’09)

If Wr(φ) has all real roots, then φ is defined over R (up to change

• f coordinates).

Challenge for geometers:

◮ M–T–V proof uses the Bethe ansatz ◮ Subsequent geometry work used M–T–V as black box. ◮ Many open generalizations of interest!

Now: conjugate roots in C and a topological approach. (−) 1 2 5 3 4 6 , (+) 1 3 5 2 4 6 , · · · Oriented Young tableaux.

Generalization: complex conjugate roots for Wr(φ)

Definition (Cutting up R[t]≤N)

For a partition µ = (2n2, 1n1), let P(µ) be P(µ) =

• polynomials with n1 distinct real roots,

n2 complex conjugate pairs

• ⊆ R[t]≤N.

Base case: µ = (1N), all real roots. Look at fibers of the restricted Wronski map: Wrµ : Wr−1(P(µ)) → P(µ).

(Note: no roots at ∞.)

Topological and algebraic degrees

How many real points in the fiber of Wrµ?

◮ Upper bound from (algebraic) degree = #SYT(

).

Topological and algebraic degrees

How many real points in the fiber of Wrµ?

◮ Upper bound from (algebraic) degree = #SYT(

).

◮ Lower bound from topological degree:

Wr−1(P(µ)) P(µ)

+ − + + +

Algebraic degree: 3 Topological degree: 1

Topological and algebraic degrees

How many real points in the fiber of Wrµ?

◮ Upper bound from (algebraic) degree = #SYT(

).

◮ Lower bound from topological degree:

Wr−1(P(µ)) P(µ)

+ − + + +

Algebraic degree: 3 Topological degree: 1

◮ We use a new “character” orientation on the Schubert cell.

Character orientation of Gr(k+1, n+1)

◮ Boundary between different P(µ)’s when roots collide:

∆ = {discr(Wr) = 0} ⊂ R[t]≤N.

+ P(µ = (1, 1)) discr = 0 P(µ = (2)) Gr(k+1, n+1) P(µ) P(µ′) Wr

Character orientation of Gr(k+1, n+1)

◮ Boundary between different P(µ)’s when roots collide:

∆ = {discr(Wr) = 0} ⊂ R[t]≤N.

+ P(µ = (1, 1)) discr = 0 P(µ = (2)) Gr(k+1, n+1) P(µ) P(µ′) Z Z Wr

◮ ∆ lifts to Z

∪ Z , two kinds of “double flex”:

◮ Type : rank deficiency at φ(k−1) rather than φ(k). ◮ Type

: rank deficiency at φ(k) and again at φ(k+1).

Character orientation of Gr(k+1, n+1)

◮ Boundary between different P(µ)’s when roots collide:

∆ = {discr(Wr) = 0} ⊂ R[t]≤N.

+ + + + + + + + + − − − P(µ = (1, 1)) discr = 0 P(µ = (2)) Gr(k+1, n+1) P(µ) P(µ′) Z Z Wr

◮ ∆ lifts to Z

∪ Z , two kinds of “double flex”:

◮ Type : rank deficiency at φ(k−1) rather than φ(k). ◮ Type

: rank deficiency at φ(k) and again at φ(k+1).

◮ Character orientation: multiply by the equation of Z .

The topological degree of Wrµ

Theorem (L, Purbhoo ‘19)

Under the character orientation, the topological degree of Wrµ is the symmetric group character χ (µ).

The topological degree of Wrµ

Theorem (L, Purbhoo ‘19)

Under the character orientation, the topological degree of Wrµ is the symmetric group character χ (µ). Murnaghan–Nakayama rule: µ = (2n2, 1n1), shape = λ: χλ(µ) =

• T

(−1)# (T) : µ-domino tableaux (+) 1 2 4

3 5 6 , (−) 1 3 4 2 5 6 , · · ·

Vertical dominos contribute (−) signs.

The topological degree of Wrµ

Theorem (L, Purbhoo ‘19)

Under the character orientation, the topological degree of Wrµ is the symmetric group character χ (µ). Murnaghan–Nakayama rule: µ = (2n2, 1n1), shape = λ: χλ(µ) =

• T

(−1)# (T) : µ-domino tableaux (+) 1 2 4

3 5 6 , (−) 1 3 4 2 5 6 , · · ·

Vertical dominos contribute (−) signs.

◮ Special case: µ = (1N), no dominos χ

(1N) = #SYT.

◮ Corollary: Shapiro–Shapiro Conjecture.

Labeling fibers by signed Young tableaux

Proof sketch (geometric Murnaghan–Nakayama rule):

◮ Use the limit fibers labeled by tableaux. ◮ Track +/− signs along boundary paths.

Labeling fibers by signed Young tableaux

Proof sketch (geometric Murnaghan–Nakayama rule):

◮ Use the limit fibers labeled by tableaux. ◮ Track +/− signs along boundary paths.

Case 1:

3 4 ← → 3 4 / 3 4 ← → 3 4

+ + + + + + + + + − − − P(µ = (16)) P(µ′ = (2, 14)) P(µ) P(µ′) Z Z 1 3 4 2 5 6 1 3 5 2 4 6 1 3 4 2 5 6 1 3 5 2 4 6

(+1) (−1)

Labeling fibers by signed Young tableaux

Proof sketch (geometric Murnaghan–Nakayama rule):

◮ Use the limit fibers labeled by tableaux. ◮ Track +/− signs along boundary paths.

Case 2:

3 4

← →

4 3

+ + + + + + + + − − P(µ = (16)) P(µ = (2, 14)) P(µ) P(µ′) Z Z 1 2 3 4 5 6 1 2 4 3 5 6 (empty)

(+1)

Open questions

◮ (Representation theory).

Do all SN character values χλ(µ) give topological degrees of real Schubert problems? (µ = (2a1b))

◮ (Complex geometry).

Explicit geometry of Wrµ over P(µ) for µ = (1N)?

◮ (Stable curves).

How does the geometry look (for µ = (1N)) over M0,N?

◮ M0,N(R) is non-orientable!

Many interesting relationships to find.

Thank you!