Promotion of increasing tableaux: frames and homomesies

A key fact about M.-P. Sch\"{u}tzenberger's (1972) promotion operator on rectangular standard Young tableaux is that iterating promotion once per entry recovers the original tableau. For tableaux with strictly increasing rows and columns, H. Thomas and A. Yong (2009) introduced a theory of $K$-jeu de taquin with applications to $K$-theoretic Schubert calculus. The author (2014) studied a $K$-promotion operator $\mathcal{P}$ derived from this theory, but showed that the key fact does not generally extend to $K$-promotion of such increasing tableaux. Here we show that the key fact holds for labels on the boundary of the rectangle. That is, for $T$ a rectanglar increasing tableau with entries bounded by $q$, we have $\mathsf{Frame}(\mathcal{P}^q(T)) = \mathsf{Frame}(T)$, where $\mathsf{Frame}(U)$ denotes the restriction of $U$ to its first and last row and column. Using this fact, we obtain a family of homomesy results on the average value of certain statistics over $K$-promotion orbits, extending a $2$-row theorem of J. Bloom, D. Saracino, and the author (2016) to arbitrary rectangular shapes.


Introduction
An important application of the theory of standard Young tableaux is to the product structure of the cohomology of Grassmannians. Much attention in the modern Schubert calculus has been devoted to the study of analogous problems in K-theory (see [PY16,§1] for a partial survey of such work). In particular, H. Thomas and A. Yong [TY09] gave a K-theoretic Littlewood-Richardson rule by developing a combinatorial theory of increasing tableaux as a K-theoretic analogue of the classical theory of standard Young tableaux. Their Littlewood-Richardson rule and the associated combinatorics has since been extended to the other minuscule flag varieties [CTY14,BR12,BS16] and into torus-equivariant K-theory [TY13,PY15].
The theory of increasing tableaux is moreover of independent combinatorial interest. Various enumerative combinatorics results have recently been obtained [Pec14,PSV16,GMP`16]; as well as applications to the studies of combinatorial Hopf algebras [PP16], longest increasing subsequences of random words [TY11], plane partitions [DPS17,HPPW16], and combinatorial representation theory [Rho17]. This paper continues the study begun in [Pec14] of the K-promotion operator on increasing tableaux, a K-theoretic analogue of M.-P. Schützenberger's [Sch72] classical promotion operator.
We systematically identify a partition λ with its Ferrers diagram in English orientation. An increasing tableau of shape λ is a filling of λ by positive integers such that entries strictly increase from left to right across rows and from top to bottom down columns. We write Inc q pλq for the set of all increasing tableaux of shape λ with entries bounded above by q. Using the K-theoretic jeu de taquin of [TY09], one has a K-promotion operator P on increasing tableaux [Pec14] by analogy with M.-P. Schützenberger's classical promotion for standard Young tableaux [Sch72]. We describe this operator in detail in Section 2.
The operation of (K-)promotion is of particular interest for tableaux of rectangular shapes. For a standard Young tableau T of shape mˆn, one has that P mn pT q " T (cf. [Hai92]); indeed, one can completely enumerate the orbits by size in this case via the cyclic sieving phenomenon [Rho10]. For increasing tableaux, on the other hand, orbits can be much larger than the cardinality q of the alphabet [Pec14, Example 3.10]. In general, no upper bound is known on the cardinality of the K-promotion orbit of a increasing tableau, even of rectangular shape. Although one might naively expect the cardinality of its K-promotion orbit to divide 26 by analogy with the standard Young tableau case, in fact the orbit of T has size 1222 " 26¨47. ♦ The frame of a partition λ is the set Framepλq of all boxes in the first or last row, or in the first or last column. For T P Inc q pλq, we write FramepT q for the labeling of T restricted to Framepλq.
Our first main result is the following: Theorem 1.2. Let T P Inc q pmˆnq. Then FramepT q " FramepP q pT qq. Note that in accordance with Theorem 1.2, every entry of FramepP 26 pT qq is bold. ♦ Remark 1.4. Since Framep2ˆnq " 2ˆn, Theorem 1.2 in particular recovers the author's previous result [Pec14, Theorem 1.3] that P q pT q " T for T P Inc q p2ˆnq.
The following was conjectured in work with K. Dilks and J. Striker [DPS17, Conjecture 4.12]: Conjecture 1.5. Let T P Inc q p3ˆnq. Then T " P q pT q.
Theorem 1.2 may be interpreted as evidence toward Conjecture 1.5, since Theorem 1.2 shows that T and P q pT q have the same entries in at least 2n`2 out of 3n pairs of corresponding boxes.
A set U of objects with a weight function wt : U Ñ C and a group action G ñ U is said to be homomesic if every G-orbit O has the same average weight ř xPO wtpxq |O| . This notion was isolated by J. Propp and T. Roby [PR15] in response to observations of D. Panyushev [Pan09], and has since been found in to appear in diverse situations [EP14, HZ15, Str15, RW15, DW16, EFG`16, JR17].
Using Theorem 1.2, we obtain our second main result, a family of new homomesies for increasing tableaux. For T P Inc q pλq and S a set of boxes in λ, let wt S pT q denote the sum of the entries of T in S.
The analogue of Theorem 1.6 for (semi)standard Young tableaux was conjectured by J. Propp and T. Roby [PR13] and proved by J. Bloom, O. Pechenik and D. Saracino [BPS16, Theorem 1.1]. In fact, for (semi)standard Young tableaux, S need not be contained in Framepmˆnq. However, [BPS16, Example 6.4] shows that for increasing tableaux a generalization of Theorem 1.6 without the condition S Ď Framepmˆnq would be false.

K-jeu de taquin and frames of increasing tableaux
The section culminates in a proof of Theorem 1.2. First we recall the K-jeu de taquin of H. Thomas and A. Yong [TY09], the key ingredient in the operation of K-promotion on increasing tableaux. While K-promotion can be defined without a full development of K-jeu de taquin, we will need K-jeu de taquin in the proof of Theorem 1.2.
2.1. K-jeu de taquin. Let BulletTableauxpν{λq denote the set of all fillings of the skew shape ν{λ by positive integers and symbols ‚. For each positive integer i, we define as follows an operator swap i on BulletTableauxpν{λq. Let T P BulletTableauxpν{λq and consider the boxes of T that contain either i or ‚. The set of such boxes decomposes into edge-connected components. On each such component that is a single box, swap i does nothing. On each nontrivial component, swap i simultaneously replaces each i by ‚ and each ‚ by i. The resulting element of BulletTableauxpν{λq is swap i pT q.
In computing swap 2 pT q, one looks at two connected components. The southwest component is a single box containing ‚ and is unchanged by swap 2 . The other component has six boxes. Hence For a box b in a partition, we write b Ñ for the box immediately right of b in its row, b Ó for the box immediately below b in its column, etc.
Consider a skew shape ν{λ. An inner corner of ν{λ is a box b P λ such that b Ñ R λ and b Ó R λ. For I any nonempty set of inner corners of ν{λ and T P Inc q pν{λq, let In I pT q be the extension of T formed by adding a ‚ to each box of I. Note that In I pT q P BulletTableauxpν{θq for some θ Ą λ.
An outer corner of ν{λ is a box b P ν{λ such that b Ñ R ν{λ and b Ó R ν{λ. If T P BulletTableauxpν{λq has all ‚'s in outer corners, then we define Out ‚ pT q to be the filling obtained by deleting every ‚ from T ; otherwise Out ‚ pT q is undefined. Note that if Out ‚ pT q is defined, then it has shape δ{λ for some δ Ď ν.
Let T P Inc q pν{λq and let I be any nonempty set of inner corners of ν{λ. Then the K-jeu de taquin slide of T at I is the result of the following composition of operations slide I pT q :" Out ‚˝s wap q˝¨¨¨˝s wap 2˝s wap 1˝I n I pT q.
Observe that slide I pT q P Inc q pδ{ρq for some ρ Ă λ and δ Ă ν.
Iterating this process for successive nonempty sets of inner corners I 1 , I 2 , . . ., one eventually obtains an increasing tableau R P Inc q pκq of some straight shape κ. Such a tableau is called a rectification of T .
Remark 2.2. Unlike in the classical standard tableau setting, an increasing tableau T P Inc q pν{λq may have more than one rectification and these rectifications may moreover have different straight shapes.
2.2. K-promotion. For T P BulletTableauxpν{λq, we define an operation Rep 1Ñ‚ that replaces each instance of 1 by ‚, as well as, for each n P Z ą0 , an operation Rep ‚Ñn that replaces each instance of ‚ by n. Let Decr be the operator that decrements each numerical entry by 1 (and ignores ‚'s).
K-promotion on Inc q pλq Ă BulletTableauxpλq is the composition P :" Decr˝Rep ‚Ñq`1˝s wap q˝¨¨¨˝s wap 3˝s wap 2˝R ep 1Ñ‚ .
It is not hard to see that if T P Inc q pλq, then PpT q P Inc q pλq, and that moreover this operation coincides with M.-P. Schützenberger's definition of promotion [Sch72] in the case that T is a standard Young tableau. For more details, see [Pec14]. Decr ♦ 2.3. K-evacuation and its dual. To prove Theorem 1.2 on K-promotion, we will need the related notion of (dual) K-evacuation. Define the shape of an increasing tableau T to be shpT q " λ if T P Inc q pλq for some q. Write T ďa for the subtableau of T given by deleting all entries greater than a and removing all empty boxes. In analogous fashion, define T ăa , T ěa , and T ąa , where T ěa and T ąa will generally be of skew shape. Note that T P Inc q pλq is uniquely determined by the vector of partitionś shpT ď0 q, shpT ď1 q, . . . , shpT ďq q¯.
For T P Inc q pλq, we define the K-evacuation of T to be the tableau EpT q encoded by the vector´s hpP q pT q ď0 q, shpP q´1 pT q ď1 q, . . . , shpP 0 pT q ďq q¯.
It is useful to encode all these data in a K-theoretic growth diagram as in [TY09], using ideas that originate in work of S. Fomin (cf. [Sta99, Appendix 1]); the K-theoretic growth diagram for T P Inc q pλq is a semi-infinite 2-dimensional array formed by placing the partition shpP j pT q ďi q in position pi`j, jq, where 0 ď i ď q and j P Z. Here the top illustrated row encodes T , the bottom row encodes P 11 pT q, and the central column encodes EpT q (which is also E˚pP 11 pT qq). ♦ Using the K-theoretic growth diagram, it is not hard to uncover various relations between the operators under consideration. Together [TY09,Theorem 4.1] and [Pec14, Lemma 3.1] give the following facts that we will need: Lemma 2.5. The following relations hold for operations on Inc q pλq: where rotpT q is given by rotating T by 180 0 and replacing i by q`1´i.

2.4.
Proof of Theorem 1.2. Let w be the reading word of T , given by reading the entries of T by rows from left to right and from bottom to top, i.e. in "reverse Arabic fashion." Let rotpwq :" w 0¨w¨w0 , where w 0 is the longest element of the symmetric group S q . Since rotpwq is obtained from w by reversing the order of the letters of w and then replacing i by q`1´i, we see that rotpwq is the reading word of rotpT q.
Define w ďa to be the subword of w obtained by deleting all letters greater than a, with analogous definitions of w ăa , w ěa , and w ąa .
Lemma 2.6. The tableaux rotpT q and EpT q have the same first row.
Proof. The reading word of rotpT q ďa is rotpwq ďa . Hence by [TY09, Theorem 6.1], the length of the first row of rotpT q ďa is LISprotpwq ďa q, where LISpuq denotes the length of the longest strictly increasing subsequence of the word u. By the definition of rot, we have LISprotpwq ďa q " LISpw ąn´a q. But by [TY09, Theorem 6.1], LISpw ąn´a q is the length of the first row of any K-rectification of T ąn´a . By definition, the shape of EpT q ďa is the shape of a particular K-rectification of T ąn´a . Thus the length of the first row of EpT q ďa is also the length of the first row of rotpT q ďa . The lemma follows.
Lemma 2.7. The tableaux rotpT q and EpT q have the same first column.
Proof. The proof is the same as for Lemma 2.6, except that one should replace use of [TY09, Theorem 6.1] on the relation between first rows and longest increasing subsequences with use of the analogous relation between first columns and longest decreasing subsequences (see [TY11] or [BS16, Corollary 6.8]).
The following proposition is of independent interest. It extends [Pec14, Proposition 3.3], which is the special case where T P Inc q p2ˆnq.
Proposition 2.8. The tableaux rotpT q and EpT q have the same frame.
Proof. By Lemmas 2.6 and 2.7, it remains to show that rotpT q and EpT q have the same last row and column.
Let T 1 " EpT q. Then by Lemmas 2.6 and 2.7, rotpT 1 q and EpT 1 q have the same first row and column. But by Lemma 2.5(a), EpT 1 q " T , so rotpT 1 q and T have the same first row and column. Hence rotprotpT 1 qq and and rotpT q have the same last row and column. Since rotprotpT 1 qq " EpT q, we are done.
Clearly rotpT q and rotprotprotpT qqq have the same frame. Since by Lemma 2.5(d), we have E˚" rot˝E˝rot, it thereby follows from Proposition 2.8 that rotpT q and E˚pT q also have the same frame. Thus E˚pEpT qq has the same frame as rotprotpT qq " T . But by Lemma 2.5(b), P q pT q " E˚pEpT qq, so FramepT q " FramepP q pT qq.
This concludes the proof of Theorem 1.2.

Homomesy
In this section, we prove a family of new homomesy results for increasing tableaux. We will obtain these by imitating the proof of [BPS16, Theorem 1.1] and using Theorem 1.2.
For T P Inc q pmˆnq and b a box in Framepmˆnq, let DistpT, bq be the multiset DistpT, bq :" twt tbu pP k pT qq : 0 ď k ă qu.
Proposition 3.1. For T P Inc q pmˆnq and b a box in Framepmˆnq, Proof. This proof is perhaps best understood by following along with the succeeding Example 3.2. Consider the K-theoretic growth diagram G for T . A fixed row r of G encodes an increasing tableau R. The row immediately below this encodes PpRq. The column that intersects r at its rightmost partition encodes, by definition, EpRq. The column immediately left of this then encodes PpEpRqq by Lemma 2.5(c). Say the rank of a partition π in G is the number rankpπq of partitions that are strictly left of π and in its row. Note that the rank is also the number of partitions strictly below π in its column.
Shade each partition in G that contains the box b. For any set of q consecutive rows tr i : 0 ă i ď qu, we have by Theorem 1.2 the equality of multisets DistpT, bq " trankpρ i q : 0 ă i ď qu, where ρ i is the leftmost shaded partition in row r i . In the same way, for any set of q consecutive columns tc j : 0 ď j ă qu, we have DistpEpT q, bq " trankpγ j q : 0 ď j ă qu, where γ j is the bottommost shaded partition in column c j .
Fix C P Z. For 1 ď k ď 2q`1, let d k denote the diagonal line of slope one through G given by y " x´k`C. For each diagonal d k , let δ k be the smallest shaded partition that lies on d k . Observe that the partitions δ k are restricted to q`1 consecutive rows tr i : 0 ď i ď qu of G and to q`1 consecutive columns tc j : 0 ď j ď qu of G. We have the equalities of multisets trankpρ i q : 0 ă i ď qu " trankpδ k q : 1 ă k ď 2q`1, rankpδ k q " rankpδ k´1 q´1u and trankpγ j q : 0 ď j ă qu " trankpδ k q : 1 ď k ă 2q`1, rankpδ k q " rankpδ k`1 q´1u. Now construct a lattice path P in the first quadrant of the plane by plotting the points pk, rankpδ k qq for 1 ď k ď 2q`1 and connecting the vertex pk, rankpδ k qq to the vertex pk`1, rankpδ k`1 qq by a line segment. Note that rankpδ 1 q " rankpδ 2q`1 q by Theorem 1.2. Moreover, rankpδ k`1 q " rankpδ k q˘1 for all k. Hence for any positive integer h, the number of k in the interval r2, 2q`1s with rankpδ k q " h " rankpδ k´1 q´1 equals the number of k in the interval r1, 2qs with rankpδ k q " h " rankpδ k`1 q´1. This proves that trankpρ i q : 0 ă i ď qu " trankpγ j q : 0 ď j ă qu and the proposition follows.  where we have also labeled 23 consecutive diagonals. We then plot the following lattice path. For O the K-promotion orbit of T , we have by Theorem 1.2 that ř U PO wt tbu pUq |O| " ř q´1 i"0 wt tbu pP i pT qq q .
Finally by Proposition 2.8, ř q´1 i"0 wt tbu pEpP i pT qqq q " ř q´1 i"0 wt tbu protpP i pT qqq q " ř q´1 i"0`q`1´w t tb˚u pP i pT qqq , where b˚is the image of b under rotating mˆn by 180 0 .
Hence putting these facts together, we have ř U PO wt tb,b˚u pUq |O| " ř q´1 i"0`q`1´w t tb˚u pP i pT qqq`ř q´1 i"0 wt tb˚u pP i pT qq q " ř q´1 i"0 pq`1q q " q`1.
Thus for S any set of boxes in Framepmˆnq that is fixed under 180 0 rotation, we have ř U PO wt S pUq |O| " pq`1q|S| 2 , as desired.