Tuesday May 30

The hive -> honeycomb bijection.
The puzzle -> honeycomb injection, and its image.

c_lambda,mu^nu nonzero
<=> there exist integral hives with that boundary
=====> there exist real hives with that boundary
<=> that boundary satisfies all puzzle inequalities
<=> for certain smaller integer honeycombs, the boundary satisfies certain inequalities
<=> for certain smaller (lambda',mu',nu') with c_lambda',mu'^nu' nonzero, (lambda,mu,nu) satisfy certain inequalities.

This becomes a recurrence once one can reverse the long arrow, which is now called the saturation theorem. The recursion was conjectured by Horn, in the context of the sums of Hermitian matrices.

Exercises #7

1. Show that the number of rhombi pointing at one side of a puzzle is the number of inversions on that side (counting counterclockwise).

2. Break a puzzle into regions by joining two pieces if they are of the same shape and share an edge, so there are 0-regions, 1-regions, and rhombus regions. Show that each region is convex, and that rhombus regions are parallelograms.

3. Orient the edges in a puzzle that separate regions; clockwise around 0-regions, counterclockwise around 1-regions. (Erase the other edges.) Classify the possible local pictures this directed graph can take around an interior or exterior vertex.

4. Let alpha,beta be two strings of k 1s and n-k 0s, labeling the SW and SE edges of a puzzle triangle read left to right.
a) If there is a puzzle, filling in this triangle, show that the last i entries of alpha have as many or more 1s as the first i entries of beta (for each i=1..n), a sort of dominance condition.
b) Show the converse -- if this dominance holds, then there is a filling.

Thursday May 18

Every overlapping combination of rhombus inequalities with no dependence on the interior is a positive combination of nonoverlapping ones, and every nonoverlapping combination can be uniquely labeled to make a puzzle. (We didn't, and won't, prove these.)
Harder fact: the only puzzles that give important inequalities are the ones that can filled in uniquely given their boundary.
Deflation of puzzles. Consequences for the number of each kind of piece.
The puzzle ring for (n,k), and its relation to the hive ring. (We have yet to prove that the natural map is a ring homomorphism.)
Definition of honeycombs.

Exercises #6

1. There is a better RSK correspondence:
{matrices of natural numbers with row sums mu₁ and column sums mu₂}
correspond to
{pairs (A,B) of two semistandard Young tableaux of the same shape, A content mu₁ and B content mu₂}
(semistandard means weakly increasing in each row, strictly in each column).
a. Show how the RSK we defined in class is a special case.
b. Show how to derive this from the hive associator, by tensoring with Altⁱ Cⁿ at each step (for various i) rather than just Cⁿ as we did.

2. Find all the puzzles of size 3.

3. Assume that a puzzle has only one 1 on one side. Give a description of all such puzzles. Show that for each size n, there are {n+1 choose 2} of them.

4. Consider a positive linear combination of rhombus inequalities where no two rhombi overlap in a triangle. If the dependence on the interior all cancels, show that the coefficients on the rhombi are all equal (so may as well be scaled to 1).

Tuesday May 16

An application of hive associativity: the RSK correspondence, which says that
words <-> pairs consisting of a GC pattern and a standard Young tableau.

Puzzles. Each puzzle gives an inequality on the possible boundary values of a hive.

Thursday May 11

Hives define an associative ring; the octahedron recurrence provides an associator.
Henriques' theorem (statement only): the associator satisfies the pentagon identity.
The accessibility lemma from linear programming (statement only).
Consequence for hives: if we can figure out all the ways to add up rhombus functionals such that the dependence on the interior vertices cancels, we'll learn the possible boundaries of real hives.

Tuesday May 9

For Sasha Postnikov's talk:
Schur polynomials. Schur functions. Schur positivity.

Example of a tensor product calculation using hives.
Definition of the hive not-necessarily-associative ring.
Proposition: as a ring, Rep(GL(n)) = Z[the fundamental reps, and det^-1].

Exercises #5

1. Use the Steinberg rule to prove the Pieri rule.
2. Show that the hive theorem is true for the following cases:

tensoring with the trivial representation

the case V_(2,1,0)^@2

the Pieri cases.

3. A hive is defined by its convexity. Hence, if we add a linear function to a hive we get another one. What is the corresponding statement about tensor products?
4. Fix lambda,mu,nu. Show that the space of real hives with this boundary is bounded (hence compact).
5*. Fix lambda,mu,kappa. What does the Steinberg rule say about the limit of
c_{N lambda,mu}^{N lambda + kappa} as N -> infinity? (We dealt with the case that lambda is regular dominant, i.e. has no repeats.)

Thursday May 4

The Steinberg tensor product formula. The flopping-and-subtraction interpretation. Statement of the Pieri rule. Certain limits of tensor product multiplicities are weight multiplicities. Decomposing a double tensor product vs. computing the invariant space in a triple tensor product.
Definition of hives. Statement of the theorem: hives compute tensor products.