The Algebra of Schur Operators
Ricky Ini Liu, Christian Smith

TL;DR
This paper explores the algebraic structure of Schur operators acting on partitions, providing a comprehensive list of relations that define their algebraic behavior, which enhances understanding of their combinatorial and algebraic properties.
Contribution
It offers a complete characterization of the relations in the algebra generated by Schur operators, advancing the algebraic understanding of the plactic monoid representation.
Findings
Complete list of relations in the algebra of Schur operators
Clarification of how Schur operators act on partitions
Insights into the algebraic structure of the plactic monoid
Abstract
We study a representation of the (local) plactic monoid given by Schur operators , which act on partitions by adding a box in column (if possible). In particular, we give a complete list of the relations that hold in the algebra of Schur operators.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algebraic structures and combinatorial models · Advanced Topics in Algebra
The Algebra Of Schur Operators
Ricky Ini Liu
Department of Mathematics, North Carolina State University, Raleigh, NC
and
Christian Smith
Department of Mathematics, North Carolina State University, Raleigh, NC
Abstract.
We study a representation of the (local) plactic monoid given by Schur operators , which act on partitions by adding a box in column (if possible). In particular, we give a complete list of the relations that hold in the algebra of Schur operators.
R. I. Liu and C. Smith were partially supported by National Science Foundation grant DMS-1700302.
1. Introduction
The Schur operator (or column box-adding operator) for acts on partitions by adding a box to the th column of the Young diagram of if the resulting diagram is a partition, otherwise sends to [math]. These operators were introduced by Fomin in [2] and also described by Fomin and Greene in [3] in their development of the theory of noncommutative Schur functions (which are a useful tool for studying Schur positivity and related phenomena). They can also be thought of as refinements of the box-adding operator acting on Young’s lattice as defined by Stanley [6] in his study of differential posets.
The authors of [3] observe that the Schur operators satisfy the relations of the local plactic monoid/algebra with relations:
[TABLE]
However, they remark that the full set of relations satisfied by the is unknown. In this paper we describe the complete set of relations among the and thereby give a full characterization of the algebra of Schur operators by proving the following theorem.
Theorem**.**
The algebra of Schur operators is defined by the relations:
[TABLE]
Interestingly, this algebra is somewhat more complicated than a more common related one, also described in [3] (see also [1]), that is generated by diagonal box-adding operators that add a box to the th diagonal of if possible (where the diagonals are labeled from bottom to top). The algebra generated by such operators was shown in [1] to be the nil-Temperley-Lieb algebra given by the relations:
[TABLE]
We will begin with some preliminary background in Section and then move on to a proof of our main theorem in Section .
2. Preliminaries
In this section, we will introduce necessary background about partitions, Knuth equivalence, and Schur operators.
2.1. Partitions
A partition of is a nonincreasing sequence of nonnegative integers. (We may add or delete trailing zeroes as convenient.) To each partition, we associate a Young diagram, which is a collection of left aligned boxes with boxes in the first row, boxes in the second row, and so on. We also define the conjugate partition to be the partition whose Young diagram is obtained from that of by reflecting across its main diagonal.
The set of partitions forms a partially ordered set called Young’s lattice , where if and only if the Young diagram of fits inside the Young diagram of (or equivalently, for all ). In this partial order, covers if and only if is a single box. Here, denotes the skew Young diagram obtained by deleting those boxes in that are also contained in .
A semistandard Young tableau (SSYT) of shape is formed by filling each box of the Young diagram of with a positive integer such that the numbers are weakly increasing within a row (read from left to right) and strictly increasing within a column (read from top to bottom). A standard Young tableau (SYT) is a semistandard Young tableau of shape with labels .
The reading word of a tableau is the word obtained by listing the entries of the tableau by rows from bottom to top, reading each row from left to right.
\ytableausetup
centertableaux
Example 2.1**.**
Let . For the semistandard Young tableau
[TABLE]
we have .
The weight of a tableau is the tuple , where is the number of occurrences of in . We similarly define the weight of any word in the alphabet . (Clearly and have the same weight.)
2.2. Schur operators
Let be the free associative algebra (over ) generated by for . Given a word in the alphabet , we define the element . Hence the set of for all words forms a basis for .
Let be the complex vector space with basis . Then acts on as Schur operators by
[TABLE]
and , extended linearly.
Example 2.2**.**
Let . Then , , but since adding another box to the second column does not yield a partition. \ytableausetupsmalltableaux
[TABLE]
Let be the two-sided ideal of consisting of all elements that annihilate all of . Then two elements and of are equivalent modulo , written , if for all partitions . We call the algebra of Schur operators.
As mentioned in the introduction, the Schur operators were introduced by Fomin [2] and discussed by Fomin and Greene [3] in their study of noncommutative Schur functions. In particular, they observe that the Schur operators give a representation of the local plactic monoid, meaning that the following relations hold modulo :
[TABLE]
For completeness, we will verify these relations in Section 3 below.
2.3. Knuth equivalence and RSK
Consider words , in the alphabet . We say that and are Knuth equivalent, denoted , if one can be obtained from the other by applying a sequence of Knuth or plactic relations of the form
[TABLE]
Here, the ellipses indicate that the subwords occurring before and after the swapped letters remain unchanged. (The Knuth relations define the so-called plactic monoid [4], of which the local plactic monoid is a quotient.)
The Robinson-Schensted-Knuth (RSK) algorithm gives a bijection between words and pairs of tableaux where the insertion tableau is semistandard, the recording tableau is standard, and and have the same shape. (See, for instance, [5] for more information.) The exact details of the RSK algorithm will not be important for us, as we will only need the following facts.
- •
The insertion tableau has the same weight as .
- •
Two words and are Knuth equivalent if and only if they have the same insertion tableau .
- •
For any semistandard tableau , the insertion tableau of is .
For instance, these facts imply the following proposition, which we will need for our main theorem. (Here and elsewhere, we use to denote a subword of the form .)
Proposition 2.3**.**
Let be a word with minimum letter , and let . Then is Knuth equivalent to a word in which all occurrences of are consecutive.
Proof.
Let be the insertion tableau of , and let . Since and both insert to , we have , and has the desired form since all ’s appear next to each other in the first row of . ∎
3. Results
Recall that we define to be the ideal that gives the relations among the Schur operators acting on . The overall goal of this section is to show that is generated by the local plactic relations (1)–(3) and one additional type of relation (4) shown below.
[TABLE]
3.1. Equivalence of words
Our first step is to understand when for . To this end we let , where
[TABLE]
(Here, a suffix of is a trailing subword of the form , possibly empty.)
Proposition 3.1**.**
Let be a partition and a word. Then
[TABLE]
Proof.
If , then adding boxes to columns of must always yield a partition. Hence for all and suffixes ,
[TABLE]
so
[TABLE]
for all and , which implies .
Otherwise, if , then for some minimal suffix , so for some ,
[TABLE]
Corollary 3.2**.**
Let and be words. Then if and only if and .
Proof.
If and , then by Proposition 3.1.
Conversely, if , then let be a partition such that
[TABLE]
Then by Proposition 3.1, which implies . If instead , then suppose without loss of generality that for some . Choose such that , but . Then , so again . ∎
In other words, a word is determined modulo by and . We next verify that is a binomial ideal, so that Corollary 3.2 essentially determines all of the relations in .
Proposition 3.3**.**
The ideal is generated by elements of the form for words and such that and .
Proof.
Let be the ideal of generated by as described above. By Corollary 3.2, we have . Let be any element of . Then for some
[TABLE]
where for each , is a word, , and for .
Fix some weight and let be those words in for which , with ordered lexicographically. We construct a partition such that
[TABLE]
Proposition 3.1 gives , but for all since by the lexicographic ordering, for some . This then implies that which is a contradiction unless . Thus and so . ∎
We therefore need only determine relations that allow us to equate for all words with a fixed and .
Another useful fact about is that it satisfies a certain shift invariance.
Corollary 3.4**.**
Let and , and define and . Then
[TABLE]
Proof.
Since
[TABLE]
for all , the result follows by Corollary 3.2. ∎
For the rest of this section, we will let denote the ideal generated by relations (1)–(4). We first verify that these relations all lie in .
Proposition 3.5**.**
The relations (1)–(4) hold in , or equivalently, .
Proof.
By Corollary 3.4, we may take . Thus by Corollary 3.2, we need only check that and for the appropriate words on both sides of the relation. This is straightforward: for instance, for relation (4),
[TABLE]
The other relations follows similarly. ∎
In particular, we note the following relationship with Knuth equivalence.
Lemma 3.6**.**
Let and be words such that . Then .
Proof.
If and are related by a Knuth move that switches , then by (1) if or by (2) or (3) if . ∎
We next demonstrate that Knuth equivalence is sufficient to describe equivalence modulo for words in two letters and .
Proposition 3.7**.**
Let and be words in and . Then if and only if .
Proof.
We claim that the insertion tableau of is determined by and . Indeed, since is semistandard and contains only ’s and ’s, it has at most two rows, and can only appear in the first row. Then given , all that needs to be determined is the number of ’s in the first row. Since , Corollary 3.2 implies that . But this is clearly the number of ’s in the first row of (since has at least as many ’s in its first row as ’s in its second row).
We can now prove the proposition. The reverse direction follows from Proposition 3.5, so suppose . By Corollary 3.2, we have and . Hence and must have the same insertion tableau by the above claim, so . By Lemma 3.6, it then follows that . ∎
Note that we have shown that if our words only contain two consecutive letters, then only relations (2) and (3) are needed to determine equivalence modulo .
3.2. Key lemmas
When dealing with three or more letters, we will need to utilize relations (1) and (4). The following two lemmas will show the key contexts in which these relations will be used.
Denote by the subword of consisting only of the letters . For instance, if , then , and .
Lemma 3.8**.**
Let and be words in . If and , then modulo relation (1), that is, they are equivalent up to commutation relations.
Proof.
Note that and must have the same number of occurrences of . Then
[TABLE]
where is a word in for all . Then we must have that
[TABLE]
where and are both words obtained by shuffling together and . But and are both equivalent modulo relation (1) to and hence to each other. It follows that and are also equivalent modulo relation (1). ∎
The next lemma shows the key application of relation (4). For ease of notation, we will abbreviate by .
Lemma 3.9**.**
For any positive integer , we have the relations
[TABLE]
Proof.
We may assume . Then (using the relations indicated)
[TABLE]
Similarly,
[TABLE]
∎
3.3. Main result
We are now ready to prove our main theorem.
Theorem 3.10**.**
For words and , if and only if .
Proof.
The reverse direction is proven in Proposition 3.5, so we need only consider the forward direction. We will induct on , the largest letter appearing in and . The case is trivial, while the case follows from Proposition 3.7.
Assume the statement holds for words in letters (and hence for words in any consecutive letters by Corollary 3.4). Since and have the same number of ’s, we can construct a word in letters such that and . We will then show and , which will imply .
By assumption we have , and so by Corollary 3.2,
[TABLE]
By the inductive hypothesis, we then have
[TABLE]
We therefore need to show that if as in (5) and , then , and similarly for and as in (6). It suffices to check when the two sides of (5) or (6) differ by a single application of one of the relations (1)–(4).
First suppose the relation used in (5) involves at most one . This will be the case unless we are applying (3) with . Note that may not be a consecutive subword inside because there may be letters that occur in between the letters of in . However, by Lemma 3.8, since there is only one occurrence of in , we can commute these intervening letters to the left or right to get some equivalent to such that has as a consecutive subword. Replacing with in then gives a word such that and . Hence
[TABLE]
A similar argument holds if the relation used in (6) involves at most one . This will be the case unless we are applying (2) with . Hence it remains to check only these remaining two cases.
Suppose in equivalence (5) we are applying (3) with by replacing . As above, and need not appear consecutively inside and since there may be intervening letters . However, we may as above commute any such letters not appearing between the ’s to the right to get words and such that
[TABLE]
where is a word in .
By Proposition 2.3, for some words and in letters . Lemma 3.6 then gives . We then have:
[TABLE]
Similarly, if in equivalence (6) we are applying (2) with by replacing , then we may commute out any 1’s not appearing between the 2’s to get:
[TABLE]
∎
Corollary 3.11**.**
The algebra of Schur operators is defined by the relations:
[TABLE]
Proof.
By Proposition 3.3, is generated by elements of the form . Theorem 3.10 then shows that the above relations generate all such elements. ∎
Example 3.12**.**
Consider the words
[TABLE]
Using the construction described in Theorem 3.10, we consider the word
[TABLE]
Note that and differ by a single application of (3) with , but these subwords do not appear consecutively within or . As in the proof of Theorem 3.10, we can rewrite the part of and between the ’s using the Knuth moves to get the ’s in the middle so that we can then use commutations to get a consecutive subword of the form . We then use Lemma 3.9, followed by the reverse of the previous procedure:
[TABLE]
Since and differ by a relation that only involves a single , we need only use commutations before we can apply the appropriate relation (3):
[TABLE]
4. Acknowledgments
The authors would like to thank Sergey Fomin for bringing this problem to their attention and for useful conversations.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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