1. Symmetries of Hives, Generalized Littlewood-Richardson Fillings, and Invariants of Matrix Pairs over Valuation Rings. (preliminary report) Glenn D. Appleby * and Tamsen Whitehead Santa Clara University October 24, 2009 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A AAA A AA A A A A A AAAA A AA A A

2. Three Big Ideas: Hives Littlewood-Richardson Fillings TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AA A A A A A AAAA A AAA A A A Invariants of Matrices over Valuation Rings

3. …with rectangles whose length corresponds to the multiplicity of the entry. Let k ij denote the number of i’s in row j. Weakly increasing along rows Strictly increasing in columns “Word Condition”: Given an LR filling of type ( ¹, º ; ¸ ), we can replace the numbers in boxes (“content º ”)… # i’s in rows i to i+k ¸ # (i+1)’s in rows (i+1) to i+ k + 1 We say the above is a Littlewood-Richardson Filling (an “LR” filling) of type ( ¹, º ; ¸ ). Littlewood-Richardson Fillings Empty Boxes: ¹. Entire shape: ¸ º = ( #1’,s, #2’s, #3’s,…) Set: LR( ¹, º ; ¸ ) =

4. Thus, Littlewood-Richardson fillings can alternately be defined by inequalities among the {k ij }:

5. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A triangular array of real numbers satisfying the “rhombus inequalities”. Sum of Obtuse Entries ¸ Sum of Acute Entries. Hives: (Hives appear in the proof of the Saturation Conjecture, eigenvalue problems for Hermitian matrices, representation theory, etc…)

6. Such a hive is of type ( ¹ º ; ¸ ). We shall let Hive( ¹, º ; ¸ ) denote the set of all hives of this type. By the Rhombus Inequalities, differences of consecutive edge entries yield a decreasing Partition of Real Numbers: Hives: Note: the type of a hive may be defined with real-valued “partitions”! (though hives over the non-negative integers are often of particular interest.)

7. Invariants of Matrix Pairs If M 2 M r (R), then there are invertible P,Q so that: a diagonal matrix, where  1 ¸  2 ¸   r ¸ 0. PMQ -1 = D  Uniquely determined by the orbit of M We’ll write inv(M) = ¹ = ( ¹ 1, …, ¹ r ) = The Invariant Partition of M. Let M r (R) denote the (r £ r) matrices over R,of full rank. Let R denote a discrete valuation ring, with uniformisant t 2 R, so for all a 2 R, a=ut k for some unit u and a non-negative integer k.

8. Suppose M, N 2 M r (R) such that inv(M) = ¹, inv(N) = º, and inv(MN) = ¸. We shall let > denote the orbit of the pair (M,N) under the action of a triple of invertible matrices (P,Q,T) 2 GL r (R), acting as: This action preserves the invariant partition of both members of the pair, and their product. Let M R ( ¹, º ; ¸ ) denote the set of orbits of matrix pairs (M,N) such that inv(M) = ¹, inv(N) = º, and inv(MN) = ¸.

9. Hives Littlewood-Richardson Fillings TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AA A A Invariants of Matrix Pairs over Discrete Valuation Rings Hive( ¹, º ; ¸ ) LR( ¹, º ; ¸ )M R ( ¹, º ; ¸ )

10. © - 1 ({ h pq }) = {k ij } where k ij = (h j,(i-1) +h (j+1)(i) ) – (h ji +h (j+1)(i-1) ), for i < j. This is just the rhombus difference for right-slanted rhombi. Thus, © - 1 could be applied to hives of arbitrary type. determined an injective map:Pak and Vallejo (2005) where is given by: Hence, © ( { k ij } ) is a hive over the non-negative integers. In fact, © is onto the set of all such hives.

11. To prove the image © ( {k ij } ) of some LR filling is a hive, we must check the Rhombus Inequalities hold. For “vertical” rhombi, the inequality reduces to one of the Column Strictness inequalities in LR2. For “left-slanted” rhombi, the inequality reduces to one of the Word Condition inequalities of LR3. However, the Rhombus Inequality for “right-slanted” rhombi reduces to the condition k ij ¸ 0, but only in the case i < j.

12. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AA A A Hive ( ¹, º ; ¸ ) LR( ¹, º ; ¸ )M R ( ¹, º ; ¸ ) The direction ( ( ) has been known, in the case of R-modules, for some time, but we are interested in the inverse direction as well, and also more constructive methods.

13. Theorem (A., 1999): Let k 11,k 12, , k 1r k 22, , k 2r   k rr be a filling in LR( ¹, º ; ¸ )  Then set, k 22, , k 2r = N LR ( ¹, º ; ¸ ) ) M R ( ¹, º ; ¸ ) The i-th factor:

14. With N defined as above, and inv(M) =  = (  1,  2, ,  r ), inv(N) = = ( 1, 2, , r ), And inv(MN) = = ( 1, 2, , r ). then, The fact that {k ij } 2 LR( ¹, º ; ¸ ) implies the matrices have the correct invariants. Note: Matrix Realizations of LR fillings of conjugate type were first obtained by Sa and Azenhas in 1990.

15. This Construction gives us an injective map: Recent work (A., Whitehead, 2009) has led to a left inverse to the above, that is, a surjective map: However, there are inequivalent orbits in M R ( ¹, º ; ¸ ) giving rise to the same filling in LR( ¹, º ; ¸ ). The associated LR filling is only a discrete invariant of the orbit. such that

17. Theorem (A., Whitehead, 2009): Given a -generic matrix L, in the orbit of a pair (M,N), then for i < j: where {k ij } 2 LR( ¹, º ; ¸ ), and ( j - i )^ denotes removing the indicated row from the minor. The filling is an invariant of the orbit. A variation of the above formula can be used to compute the corresponding hive entry in Hive( ¹, º ; ¸ ). Denotes the order of the minor of L with the given rows and columns.

18. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AA A A Now, we can connect Hives to LR fillings, and LR fillings to Matrix Pair Invariants. …Provided everything is defined over the non-negative integers. Hive ( ¹, º ; ¸ ) LR( ¹, º ; ¸ )M R ( ¹, º ; ¸ )

19. Are these “fillings” © - 1 (Hive( ¹, º ; ¸ )) realized as an invariants of an actual objects of study? Call these “interior parts” Call these “edge parts”

20. Our original matrix results were over discrete valuation rings, leading to LR fillings over the non-negative integers. We have extended these results to a class of rings with a real-valued valuation. There are many such examples. For one, let denote a field of formal power series: with real-valued exponents ® i. We’ll suppose the set of exponents for any a 2 has no limit points, so multiplication in is well-defined.

21. With each elementwe may define the order of a as: We shall define R µ as the valuation ring: Whose units are the set: Over R, Not

22. We now work with full-rank matrix pairs (M,N) 2 M r ( ), And, as before, will study the orbits of the action Where pairs (M,N) 2 M r ( ), but (P,Q,T) 2 GL r ( R ). Essentially, everything goes over to the -valued case. All our constructions extend to this setting, including the determinantal formulas for fillings. In this sense, we can form the set of orbits, denoted by: M ( ¹, º ; ¸ ), where ¹, º, and ¸ are now partitions of real numbers.

23. That is, given an orbit of a matrix pair (M,N) in M ( ¹, º ; ¸ ), we can find GL r ( R ) invariants that allow us to define real numbers {k ij } that formally satisfy:

24. However, using these formulas, we find there is an additional property our “ ” fillings satisfy: That is, fillings {k ij } from matrix pairs over yield the same fillings as © - 1 (Hive( ¹, º ; ¸ )) under the Pak-Vallejo map: This appears to be the “right” generalization to real-valued LR fillings.

25. So, we can determine a “filling” {k ij }. Can we draw the diagram? We shall call the above a diagram of a real-valued Littlewood-Richardson filling, or an “ “ filling. Insert an “origin”, measured horizontally Begin by drawing the “base”, ¹. We measure to the right for positive parts, and to the left for negative parts. Interior parts (always non-negative) are added, moving to the right: Edge parts, when negative, move the end of the row to the left. The partition ¸ is measured from the origin to the end of the row (here, positive, though a negative row is possible).

26. Hive( ¹, º ; ¸ ) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AA A A So now, over, we can connect:

27. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AA A A I. From Hives, to Matrices and LR Fillings Hive( ¹, º ; ¸ )

28. Note that, if inv(M) = ¹, then We will depict a hive of this type by:

29. Inv(M) Inv( ¤ ) Inv(N) MN= ¤ N ¤ - 1 = M -1 ¤ - 1 M = N -1 ¤ N -1 = M N -1 M -1 = ¤ - 1 M -1 ¤ = N Set ¤ = MN, so that

30. So, with six different orientations of hives… come six different, but related LR Diagrams. The fillings in these six diagrams are linearly related by the S 3 symmetries of the (linearly related) entries of the corresponding hives.

31. In classical LR fillings, proving c ¹ º ¸ = c º, ¹ ¸ is established by constructing a bijection LR( ¹, º ; ¸ ), LR( º, ¹ ; ¸ ) as found in Kerber and James, and generalized by the row-switching algorithms of Benkhart, Sottile, and Stroomer (1996). What does this relationship among LR fillings suggest about matrix invariants and hives? II. From LR Fillings to Matrices and Hives

32. Recall we calculate an LR filling from matrices by using the order of a minor of the L: Omitted rows {k ij } 2 LR ( ¹, º ; ¸ ) {m ij } 2 LR ( º, ¹ ; ¸ ) L =

33. That is, we may use a -generic matrix L, to get two LR fillings, the first in LR( ¹, º ; ¸ ) as before, and a second in LR( º, ¹ ; ¸ ): (M,N)(M,N) { k ij } 2 LR( ¹, º ; ¸ ){ m ij } 2 LR( º, ¹ ; ¸ ) (“left filling”)(“right filling”) Theorem (A., Whitehead, 2009): Given any matrix pair (M,N), and any right filling associated to the orbit of the pair, there is a unique left filling also associated to the pair that is independent of the matrix realization of the filling. Further, if ¹, º, and ¸, along with the filling, are realized by non- negative integers, this algebraic bijection of fillings is the same as the combinatorial bijection described by Kerber and James for classical LR fillings. “content º ”“content ¹ ”

34. Theorem (A., Whitehead, 2009) : In terms of hives, this bijection on fillings and matrix pairs: into one of the form: Note that, in particular, this new hive is not obtainable by any of the S 3 symmetries found earlier. yields a new involution on hives, (which we can describe independently of the filling) taking each hive of the form:

35. III. From Matrix Pairs to LR Fillings and Hives There’s a nice description of the scalar shift of a hive, too! (Left) LR Filling:(Right) LR Filling: Scalar shift on the right only shifts the edge parts of the Right Filling The Left Filling is unchanged, though the origin is shifted. For scalar shifts of the left member, these phenomena are reversed. – Interior parts are invariant under scalar shifts. Content º “Base” º

36. (M,N)(M,N) LR Filling of ¸ / ¹ with content º LR Filling of ¸ / º with content ¹ (“left filling”)(“right filling”) …where both sides are related by the “left-right” LR bijections. Then, by the hive symmetries, we obtain the related fillings: Our interest is not so much in counting fillings of a given type, but to understand the interconnections between seemingly different fillings, and the objects that relate them.

37. (M,N)(M,N) And, for each family of fillings, we obtain a new family (with the same interior parts) by various scalar shifts: And, for each family of fillings, Etc….

38. These results imply, among other things, that huge families of scalar-shifted, “left-right”-related, or hive-symmetry- related triples of partitions have equal LR coefficients, in that they all share the same “combinatorial core”... …and there’s more... …but time’s up! Thank you!

39. What is the Hive corresponding to the scalar shift

40. A shift of the left member corresponds to a adding a “staircase” down the hive : Pleasant Exercise: Calculate the hive and diagram for:

41. Given the Hive: We first depict the (non-negative) partition º : Where: We then extend the diagram by the non-negative interior parts k ij, of the filling where i < j… …followed by the (negative) edge parts k ii : Which results in the shape.