Lascoux and Schützenberger  introduced the plactic monoid to give a proof of the Littlewood-Richardson rule based on a strategy outlined by Robinson , and ideas of Schensted  and Knuth . This theory revolves around bijections involving words and tableaux.
These bijections are restrictions to lattice points of certain volume-preserving piecewise linear bijections between convex polyhedra [7, 6, 3, 14]. The importance of this viewpoint is borne out in the work of Knutson and Tao, who proved the saturation of the monoid of triples of integer partitions such that the representation
occurs in the tensor productof representations of . This led to the resolution of Horn’s conjecture
on the possible sets of eigenvalues of a sum of Hermitian matrices[9, 10].
Here we develop the monoid-theoretic foundations for piecewise linear correspondences interpolating bijections involving tableaux. This is done by generalizing the plactic monoid from the framework of the free monoid of words to the setting oftimed words. Timed words were introduced by Alur and Dill  in their approach to the formal verification of real-time systems using timed automata. While words represent a sequence of events, timed words represent a sequence of events where the time of occurrence of each event is also recorded. We use a finite version of their definition of timed words. While each letter occurs discretely (an integer number of times) in a classical word, it appears for a positive duration (which is a real number) in a timed word.
In Section 2.1 we introduce the timed versions of semistandard Young tableaux (called timed tableaux). Schensted’s insertion algorithm is generalized to timed tableaux in Section 2.2. Greene invariants of timed words are introduced in Section 3.1. Timed versions of Knuth relations are introduced in Section 3.2.1. These relations are conceptually very similar to the relations introduced by Knuth in . However, it was a delicate task to arrive at relations that are at once simple enough so that one can show that they preserve Greene invariants (Lemma 3.3.1) but also powerful enough to transform any timed word to its insertion tableau (Lemma 3.2.4). With this groundwork, the extension of Greene’s theorem to timed words becomes routine (Theorem 3.4.1).
Standard properties of Knuth equivalence, such as the existence of a unique tableau in each Knuth class, and the characterization of Knuth equivalence in terms of Greene invariants are extended to timed words in Sections 4.1 and 4.2.
The RSK algorithm  is extended from integer matrices in Sections 5.1 and 5.2. Viennot’s light-and-shadows version of the Robinson-Schensted correspondence, which was extended to integer matrices in , is now extended to real matrices. The piecewise linear nature of these algorithms can be easily seen from the timed version of Greene’s theorem.
All the algorithms described here are straightforward to implement. An implementation in python is available at http://www.imsc.res.in/~amri/timed_plactic/timed_tableau.py. A jupyter worksheet with demos of many of the theorems and proofs in this paper is available at http://www.imsc.res.in/~amri/timed_plactic/timed_tableau.ipynb and in html format at http://www.imsc.res.in/~amri/timed_plactic/timed_tableau.html.
2. Insertion in Timed Tableaux
2.1. Timed Tableaux
Let , to be thought of as a linearly ordered alphabet.
Definition 2.1.1 (Timed Word).
A timed word of length in the alphabet is a piecewise-constant right-continuous function with finitely many discontinuities. We write . In other words, for some finite sequence of transition points, and letters in , if . Given such a function, we write
where . We call this an exponential string for . The weight of
is the vector:
where is the Lebesgue measure of the pre-image of under , in other words,
The exponential string, as defined above, is not unique; if two successive letters and are equal, then we can merge them, replacing .
The above definition is a finite variant of Definition 3.1 of Alur and Dill , where , and there is an infinite increasing sequence of transition points.
Given timed words and , their concatenation is defined in the most obvious manner—their exponential strings are concatenated (and if necessary, successive equal values merged). The monoid formed by all timed words in an alphabet , with product defined by concatenation, is denoted by . The submonoid of consisting of timed words where the exponents in exponential string (1) are integers is the free monoid of words in the alphabet (see e.g., [12, Chapter 1]).
Definition 2.1.2 (Timed Subword).
Given a timed word , and a finite disjoint union of intervals of the form , the timed subword of with respect to is defined by:
Intuitively, is obtained from by cutting out the segments that are outside . Given timed words and , is said to be a timed subword of if there exists as above such that . Timed subwords of are said to be pairwise disjoint if there exist pairwise disjoint subsets as above such that for .
Definition 2.1.3 (Timed Row).
A timed row in the alphabet is a weakly increasing timed word in . In exponential notation every timed row is of the form where for . The set of all timed rows in is denoted . The set of all timed rows of length is denoted .
Definition 2.1.4 (Row Decomposition).
Every timed word has a unique decomposition:
such that is a timed row for each , and is not a row for any . We shall refer to such a decomposition as the row decomposition of .
Given two timed rows and , say that is dominated by (denoted ) if
for all .
Definition 2.1.5 (Real Partition).
A real partition is a weakly decreasing finite sequence of non-negative real numbers. Two real partitions are regarded as equal if one may be obtained from the other by the removal of trailing zeroes.
Definition 2.1.6 (Timed Tableau).
A timed tableau in is a timed word in with row decomposition such that . The shape of is the real partition , and the weight of is the weight of as a timed word (see Definition 2.1.1). The set of all timed tableaux in is denoted . The set of all timed tableaux of shape is denoted . The set of all timed tableaux of shape and weight is denoted .
The above is a direct generalization of the notion of the reading word of a tableau in the sense of .
is a timed tableau in of shape and weight .
2.2. Timed Insertion
Given a timed word and , according to Definition 2.1.2, is the timed word of length such that:
Definition 2.2.1 (Timed row insertion).
Given a timed row , a letter , and a real number , the insertion of into is defined as follows: if for all , then
where denotes the empty word of length zero. Otherwise, there exists such that . Let
When and lies in the image of in , this coincides with the first step of the algorithm INSERT from Knuth  where is inserted into a row: if , then is the new row obtained after insertion, is the letter bumped out by .
If is a row of the form , define by induction on as follows: Having defined , let . Then define
The insertion of the timed row into is achieved by successively inserting as runs from to . The segments are ejected are taken from from left to right, with no overlaps. It follows that, if , then is again a timed row.
Definition 2.2.1 is in terms of exponential strings, which are not uniquely associated to timed words. Therefore, in order that row insertion be well-defined for timed words, it is necessary to check that the result of row insertion of is the same as the result of row insertion of followed by row insertion of . This is straightforward.
Definition 2.2.5 (Timed Tableau Insertion).
Let be a timed tableau with row decomposition , and let be a timed row. Then , the insertion of into , is defined as follows: first is inserted into . If , then is inserted into ; if , then is inserted in , and so on. This process continues, generating and . is defined to be . It is quite possible that .
If is the timed tableau from Example 2.1.7, then
When the timed tableau lies in the image of , then is the same as the result of applying the algorithm from Knuth  to the semistandard Young tableau .
Given real partitions and , we say that interleaves if the inequalities
hold. In other words, the successive parts of lie in-between the successive parts of .
For any timed tableau in and any timed row in , is again a timed tableau in . We have
and interleaves .
The main step in this proof is the following lemma:
Suppose and are timed rows in such that . For any timed row in , suppose , and .
Proof of the lemma.
By inserting in stages, assume that for some and some .
If never exceeds , then , and , and so . Otherwise, when is inserted into , is a segment of corresponding to an interval such that . This segment in is replaced by a segment to obtain . Let , with , be the exponential string of .
Proceed by induction on . If , . Now , so will displace a segment of , one that begins to the left of , with , and so .
For , perform the insertion of into in two steps, first inserting , and then inserting . If , then . Let . By the case, .
Now , so is obtained by inserting into . Therefore, by induction hypothesis, , proving (1).
The assertion (2) about weights is straightforward. For (3), observe that is a concatenation of segments from . Write , where consists of segments that come from and consists of segments that come from . Then, from the arguments in the proof of the first part of the lemma, the segments in will all replace segments of , so if , then . Now , whence . ∎
We now prove that is a timed tableau. Suppose has row decomposition . Using the notation of Definition 2.2.5, and writing for , we have and . The first assertion of Lemma 2.2.9, with , , and , shows that for . Taking and gives . Therefore is a timed tableau. The third assertion of Lemma 2.2.9, with the same settings, gives , showing that interleaves . The assertion about weights in the theorem follows easily from the second assertion of Lemma 2.2.9. ∎
Definition 2.2.10 (Insertion Tableau of a Timed Word).
Let be a timed word with row decomposition . The insertion tableau of is defined as:
If has four rows in its row decomposition. is calculated via the following steps:
Definition 2.2.12 (Schützenberger Involution on Timed Words).
Given , define
in effect, reversing both the order on the alphabet, and the positional order of letters in the timed word.
Let and be timed rows. Suppose , and . Then .
It suffices to consider the case where . The hypothesis implies that satisfies , and
Using induction as in the proof of Theorem 2.2.8, we may assume that is constant, so for some .
Since all the values of are greater than or equal to , all the values of are less than or equal to . Moreover, . It follows immediately from Definition 2.2.1 that . ∎
The timed row insertion algorithm gives rise to a bijection:
Suppose , and . Then can be recovered from (given the prior knowledge of and ) as follows: let . Then using Lemma 2.2.13, and can be recovered as , and . ∎
Theorem 2.2.15 (Timed Pieri Rule).
The timed insertion algorithm gives rise to a bijection:
Let . Let have row decomposition , and . Suppose that has row decomposition (with the possibility that ). We already know that interleaves (Theorem 2.2.8). Given timed rows and such that , and non-negative real numbers and such that , let denote the unique pair of rows such that , , and (see Corollary 2.2.14). Then the rows of can be recovered from as follows:
and finally can be recovered as . ∎
Definition 2.2.16 (Deletion).
Let and let be a real partition that interleaves . Then we write
The pair is computed from and by the algorithm described in the proof of Theorem 2.2.15.
3. Greene’s Theorem
3.1. Greene Invariants for Timed Words
Definition 3.1.1 (Greene Invariants for Timed Words).
Given , its th Greene invariant is defined as the maximum possible sum of lengths of a set of pairwise disjoint subwords of (see Definition 2.1.2) that are all timed rows:
If is a timed tableau of shape , then for each ,
Conversely, any row subword of cannot have overlapping segments from two different rows and of , because if , then , but in the row decomposition of , occurs before . Therefore, disjoint subwords can have length at most the sum of lengths of the largest rows of , which is . ∎
3.2. Timed Knuth Equivalence and the Timed Plactic Monoid
Definition 3.2.1 (Timed Knuth Relations).
Assume that , and are timed rows such that is also a timed row. The timed Knuth relations are given by:
Definition 3.2.2 (Timed Plactic Monoid).
In other words, two elements of are said to differ by a Knuth relation if they are of the form and , where and are terms on opposite sides of one of the timed Knuth relations () and (). Knuth equivalence is the equivalence relation generated by Knuth relations. Since this equivalence is stable under left and right multiplication in , the concatenation product on descends to a product on the set of Knuth equivalence classes, giving it the structure of a monoid.
Two timed words and differ by a Knuth relation () if and only if and (see Definition 2.2.12) differ by a Knuth relation ().
Every timed word is Knuth equivalent to its timed insertion tableau.
First we show that, for timed rows and , if , then . By insertion in stages, we assume that . If for all , there is nothing to show. Otherwise, a segment of , beginning at , and of length is displaced by the segment of . Write . It suffices to show . But this can be done in two steps as follows (the segment to which the Knuth relation is applied is underlined):
by repeated application of the assertion at the beginning of this proof. ∎
3.3. Knuth Equivalence and Greene Invariants
If two timed words are Knuth equivalent, then they have the same Greene invariants.
It suffices to prove that if two words differ by a Knuth relation they have the same Greene invariants. For the Knuth relation (), suppose that is a timed row with , and the last letter of is strictly less than the first letter of . For any timed words and , we wish to show that Greene invariants coincide for and . Since every timed row subword of is also a timed row subword of , for all .
To prove the reverse inequality, for any set of pairwise disjoint row subwords of , it suffices to construct pairwise disjoint row subwords of such that . Write for each , where and are (possibly empty) row subwords of and respectively.
Since the last letter of is strictly smaller than the first letter of , it cannot be that and simultaneously for the same . If, for all , , or , then each remains a row subword of , so we may take for all .
Otherwise, there exists such that , with , and . Without loss of generality, assume that this is the case for , and not for for some . If for all , then set
Since , . By construction the words are pairwise disjoint timed row subwords of .
Finally, suppose there exists at least one index such that , say . Also, assume that (the largest letter of ) is at least as large as for . Let be the timed row obtained by concatenating all the segments of for in the order in which they occur in . It follows that , and . Let be the timed row obtained by concatenating all the segments of and in the order in which they occur in . If , define
If , then interchange and in the above definition. In both cases the words are pairwise disjoint row subwords of whose lengths add up to .
For the Knuth relation (), a similar argument can be given. However, a more elegant method is to use Lemma 3.2.3, noting that for all and all , thereby reducing it to (). ∎
3.4. The timed version of Greene’ theorem
Theorem 3.4.1 (Timed version of Greene’s theorem).
For every , if the timed tableau has shape , then
Greene’s theorem holds when is a timed tableau (Lemma 3.1.2). By Lemma 3.3.1, Greene invariants remain unchanged under the timed versions of Knuth relations. By Lemma 3.2.4, every timed word is Knuth equivalent to its timed insertion tableau. Therefore, the Greene invariants of a timed word are given by the shape of its insertion tableau as stated in the theorem. ∎
The proof of Lemma 3.1.2 shows that the supremum in the definition of Greene invariants (Definition