RULES OF THREE FOR COMMUTATION RELATIONS

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1 RULES OF THREE FOR COMMUTATION RELATIONS JONAH BLASIAK AND SERGEY FOMIN Dedicated to Efim Zelmanov on his 60th irthday Astract. We study the phenomenon in which commutation relations for sequences of elements in a ring are implied y similar relations for susequences involving at most three indices at a time. What I tell you three times is true. The Hunting of the Snark, y Lewis Carroll Introduction In this paper, we investigate the following surprisingly widespread phenomenon which we call The Rule of Three: in order for a particular kind of commutation relation to hold for susequences of elements of a ring laeled y any suset of indices, it is enough that these relations hold for susets of size one, two, and three. Here is a typical Rule of Three statement. Let g 1,...,g N,h 1,...,h N e invertile elements in an associative ring. Then the following are equivalent (cf. Theorem 3.4): for any susequence of indices 1 s 1 < < s m N, the element g sm g s1 commutes with oth h sm h s1 and h sm + +h s1 ; the aove condition holds for all susequences of length m 3. We estalish many results of this form, including Rules of Three for noncommutative elementary symmetric functions (Section 1); Rules of Three for generating functions over rings (Section 2); Rules of Three for sums and products (Section 3). Proofs are given in Sections 4 8. For reference, Theorems 2.5 and 2.12 are proved in Section 5; Theorem 3.2 is proved in Section 6; Theorems 1.1, 3.4, 3.5, 3.10, and 3.11 are proved in Section 7; and Theorem 1.6 is proved in Section 8. Date: August 16, Mathematics Suject Classification. Primary 16S99. Secondary 05E05. Key words and phrases. Noncommutative symmetric function, commutation relations, elementary symmetric function, Rule of Three, Dehn diagram. Partially supported y NSF grants DMS (J. B.) and DMS (S. F.). 1

2 2 JONAH BLASIAK AND SERGEY FOMIN 1. Rules of Three for noncommutative symmetric functions Let u = (u 1,...,u N ) e an ordered N-tuple of elements in a ring R. (We informally view u 1,...,u N as noncommuting variales. ) For an integer k, the noncommutative elementary symmetric function e k (u) R is defined y e k (u) = u i1 u i2 u ik. (1.1) N i 1 >i 2 > >i k 1 (By convention, e 0 (u)=1 and e k (u)=0 if k < 0 or k > N.) More generally, for a suset S {1,...,N}, we denote e k (u S ) = u i1 u i2 u ik. (1.2) i 1 >i 2 > >i k i 1,...,i k S Again, e 0 (u S )=1, and e k (u S )=0 unless 0 k S. (Here S is the cardinality of S.) Theorem 1.1 (The Rule of Three for noncommutative elementary symmetric functions). Let u = (u 1,...,u N ) and v = (v 1,...,v N ) e ordered N-tuples of elements in a ring R. Then the following conditions are equivalent: the noncommutative elementary symmetric functions e k (u S ) and e l (v S ) commute with each other, for any integers k and l and any suset S {1,...,N}: e k (u S )e l (v S ) = e l (v S )e k (u S ); (1.3) the commutation relation (1.3) holds for S 3 and any k,l; the commutation relation (1.3) holds for S 3 and kl 3. Remark 1.2. Explicitly, Theorem 1.1 asserts that the commutation relations (1.3) hold for all k and l and all susets S {1,...,N} if and only if the following relations hold: e 1 (u S )e 1 (v S ) = e 1 (v S )e 1 (u S ) for 1 S 3, (1.4) e 2 (u S )e 1 (v S ) = e 1 (v S )e 2 (u S ) for 2 S 3, (1.5) e 1 (u S )e 2 (v S ) = e 2 (v S )e 1 (u S ) for 2 S 3, (1.6) e 3 (u S )e 1 (v S ) = e 1 (v S )e 3 (u S ) for S = 3, (1.7) e 1 (u S )e 3 (v S ) = e 3 (v S )e 1 (u S ) for S = 3. (1.8) (Actually, it suffices to require (1.4) for S 2, ut this is not so important.) It is rather miraculous that (1.4) (1.8) imply the relations e 2 (u S )e 2 (v S ) = e 2 (v S )e 2 (u S ) for 2 S 3, (1.9) e 2 (u S )e 3 (v S ) = e 3 (v S )e 2 (u S ) for S = 3, (1.10) e 3 (u S )e 2 (v S ) = e 2 (v S )e 3 (u S ) for S = 3, (1.11) e 3 (u S )e 3 (v S ) = e 3 (v S )e 3 (u S ) for S = 3, (1.12) in addition to all relations (1.3) for S 4. In the case of a single N-tuple of noncommuting variales u 1 = v 1,...,u N = v N, we otain the following corollary.

3 RULES OF THREE FOR COMMUTATION RELATIONS 3 Corollary 1.3 ([11], [6]). Let R e a ring, and let u=(u 1,...,u N ) e an ordered N-tuple of elements of R. Then the following are equivalent: the noncommutative elementary symmetric functions e k (u S ) and e l (u S ) commute with each other, for any integers k and l and any suset S {1,...,N}: the following special cases of (1.13) hold: e k (u S )e l (u S ) = e l (u S )e k (u S ); (1.13) e 1 (u S )e 2 (u S ) = e 2 (u S )e 1 (u S ) for 2 S 3, (1.14) e 1 (u S )e 3 (u S ) = e 3 (u S )e 1 (u S ) for S = 3. (1.15) Proof. This is a special case of Theorem 1.1. Note that when u = v, the condition (1.4) is trivial, whereas (1.5) (1.8) ecome (1.14) (1.15). Remark 1.4. Corollary 1.3 is equivalent to a result y A. N. Kirillov [11, Theorem 2.24]; the aove version appeared in our previous paper [6, Theorem 2.3]. It generalizes similar results otained in [4, 10, 13, 15, 19]. See [6, Remark 2.2] for additional discussion. Remark 1.5. Corollary 1.3 and similar results serve as the starting point for the theory of noncommutative Schur functions, which aims to produce positive cominatorial formulae for Schur expansions of various classes of symmetric functions. This theory originated in[10], uilding off the work of Lascoux and Schützenerger on the plactic algera[15, 21]. It was later adapted to study LLT polynomials [13] and k-schur functions [12]; other variations appeared in [2, 11, 19]. Further recent work includes the papers [4, 5, 6], which advance the theory to encompass Lam s work [13] and incorporate ideas of Assaf [1]. One of the main outcomes of this approach is a proof of Haglund s conjecture on 3-column Macdonald polynomials [5]. Theorem 1.1 can e generalized to the setting of noncommutative supersymmetric polynomials. Let usfixanaritrarypartitionoftheorderedalphaet {1 < < N} into unarred and arred indices. The noncommutative super elementary symmetric function e k (u) is defined y the following variation of (1.1): e k (u) = u i1 u. (1.16) ik N i 1 i 2 i k 1 i j unarred i j > i j+1 We similarly define the elements e k (u S ) associated to su-alphaets S {1 < < N}. Theorem 1.6. Let R e a ring, and let u=(u 1,...,u N ) and v=(v 1,...,v N ) e ordered N-tuples of elements of R. Then the following are equivalent: e k (u S ) and e l (v S ) commute, for any k and l and any suset S {1,...,N}: the following special cases of (1.17) hold: e k (u S )e l (v S ) = e l (u S )e k (v S ); (1.17) e k (u S )e 1 (v S ) = e 1 (u S )e k (v S ) for S 3 and k 1; (1.18) e 1 (u S )e l (v S ) = e l (u S )e 1 (v S ) for S 3 and l 1. (1.19) We discuss the roader context for Theorem 1.6 in Remark 2.7 elow.

4 4 JONAH BLASIAK AND SERGEY FOMIN 2. Rules of Three for generating functions over rings Commutation relations for noncommutative elementary symmetric functions can e reformulated as multiplicative identities for certain elements of a polynomial ring in two (central) variales with coefficients in R. This leads to an alternative perspective on The Rules of Three, which we discuss next. In the rest of this paper, we repeatedly make use of the following convenient notation. Letg 1,...,g N eelementsofaring(oramonoid), andlets = {s 1 < <s m } {1,...,N} e a suset of indices. We then denote We similarly use the shorthand h S = h sm h s1, etc. g S = g sm g s1. (2.1) Corollary 2.1. Let R[x,y] e the ring of polynomials in the formal variales x and y with coefficients in a ring R. (Here x and y commute with each other and with any z R.) Let u 1,...,u N,v 1,...,v N R. For i = 1,...,N, set Then the following are equivalent: g i = 1+xu i R[x,y], (2.2) h i = 1+yv i R[x,y]. (2.3) g S h S = h S g S for all susets S {1,...,N}; g S h S = h S g S for all susets S of cardinality 1, 2, and 3. Proof. We oserve that (2.2) (2.3) imply g S = (1+xu sm ) (1+xu s1 ) = k xk e k (u S ), (2.4) h S = (1+yv sm ) (1+yv s1 ) = l yl e l (v S ). (2.5) Thus the property (1.3) (that is, each e k (u S ) commutes with each e l (v S )) is equivalent to saying that g S commutes with h S. The corollary is now immediate from Theorem 1.1. Given the multiplicative form of the conditions g S h S = h S g S in Corollary 2.1, it is tempting to seek group-theoretic generalizations of the latter, with the factors g i and h i drawn from some (reasonaly general) group. Unfortunately, the purely group-theoretic extension of Corollary 2.1 is false: the relation g 4 g 3 g 2 g 1 h 4 h 3 h 2 h 1 = h 4 h 3 h 2 h 1 g 4 g 3 g 2 g 1 does not hold in the group with presentation given y generators g 1,...,g 4,h 1,...,h 4 and relations g S h S = h S g S for S 3. (This follows from the fact that replacing (2.3) with h i = 1+xv i transforms Corollary 2.1 into a false statement, cf. Example 2.11.) Consequently one has to introduce some (likely nontrivial) assumptions on the group G and/or theelements g i,h i. Two results ofthis kind arestated insection 4. Afundamental question remains (see also Prolem 2.9): Prolem 2.2. Find a group-theoretic Rule of Three strong enough to directly imply Corollary 2.1 (or etter yet, Conjecture 2.3 elow).

5 RULES OF THREE FOR COMMUTATION RELATIONS 5 From the standpoint of potential applications, the most important setting for multiplicative rules of three à la Corollary 2.1 is the one where the factors g i,h i are formal power series in xu i and yv i, respectively. Extensive computational evidence suggests that in this setting, the Rule of Three always holds: Conjecture 2.3. Let R[[x,y]] e the ring of formal power series in the variales x and y with coefficients in a Q-algera R. Let u 1,...,u N,v 1,...,v N R, and assume that g 1,...,g N, h 1,...,h N R[[x,y]] are power series of the form g i = 1+α i1 xu i +α i2 (xu i ) 2 +α i3 (xu i ) 3 + (α ik Q), (2.6) h i = 1+β i1 yv i +β i2 (yv i ) 2 +β i3 (yv i ) 3 + (β ik Q), (2.7) where for every i, either α i1 or β i1 is nonzero. Then the following are equivalent: g S h S = h S g S for all susets S {1,...,N}; g S h S = h S g S for all susets S of cardinality 1, 2, and 3. While Conjecture 2.3 remains open, we were ale to prove it in several important cases. Three such results appear elow and two more appear in Corollaries 7.6 and 7.7. First, we otain the following generalization of Corollary 2.1. Theorem 2.4. Conjecture 2.3 holds for h i = 1+yv i (and any g i as in (2.6)). Theorem 2.4 is a special case of a more general result, see Corollary 7.6. We also settle Conjecture 2.3 in the case of a single set of noncommuting variales: Theorem 2.5. Let R[[x,y]] e the ring of formal power series in x and y with coefficients in a Q-algera R. Let u 1,...,u N R. Let g 1,...,g N,h 1,...,h N R[[x,y]] e of the form g i = 1+α i1 xu i +α i2 (xu i ) 2 +α i3 (xu i ) 3 + (α ik Q), (2.8) h i = 1+β i1 yu i +β i2 (yu i ) 2 +β i3 (yu i ) 3 + (β ik Q), (2.9) where for every i, either α i1 or β i1 is nonzero. Then the following are equivalent: g S h S = h S g S for all susets S {1,...,N}; g S h S = h S g S for all susets S of cardinality 2 and 3. Yet another case of Conjecture 2.3 follows from Theorem 1.6: Corollary 2.6. Conjecture 2.3 holds provided for each i, one of the following two options is chosen: g i = 1+xu i and h i = 1+yv i ; or g i = (1 xu i ) 1 and h i = (1 yv i ) 1. Theorem 1.6 is stronger than Corollary 2.6 since the latter requires the relations (1.17) for S 3 and any k,l whereas the former only needs the instances with k = 1 or l = 1.

6 6 JONAH BLASIAK AND SERGEY FOMIN Remark 2.7. Theorem 2.5 is a far-reaching generalization of Corollary 1.3, which already demonstrated its importance in initiating new developments in the theory of noncommutative Schur functions (cf. Remark 1.5). It remains to e seen whether Theorem 2.5 for general power series (2.8) (2.9) can spawn new versions of this theory. One case(eyond the asic choice(2.2) (2.3)) where some progress has een made is the setting of noncommutative super symmetric functions (cf. Theorem 1.6). The ring defined y the relations (1.14) (1.15) has many quotients with rich cominatorial structure (the plactic algera, nilcoxeter algera, and more, see [4, 6, 10, 13]); the ring defined y the relations (1.18) (1.19) has many interesting quotients as well, some of them similar to the plactic algera. The recent paper [7] studies some of these quotients and develops an accompanying theory of noncommutative super Schur functions. The main application (recovering results of [3, 16]) is a positive cominatorial rule for the Kronecker coefficients where one of the shapes is a hook. We next discuss some of the sutleties involved in generalizing the aove results. To facilitate this discussion, we introduce the following concept. Definition 2.8. Let A = Q u 1,...,u M,v 1,...,v M ethefreeassociative Q-algerawith generators u 1,...,u M,v 1,...,v M. Let g 1,...,g N,h 1,...,h N e elements of A[[x,y]], i.e., some formal power series in x and y with coefficients in A. We say that the Multiplicative Rule of Three holds for g 1,...,g N,h 1,...,h N if for any quotient ring R = A/I, the following are equivalent: g S h S h S g S mod I for all susets S {1,...,N}; g S h S h S g S mod I for all susets S of cardinality 3. For example, the Multiplicative Rule of Three holds in the following cases: h i = 1+yv i and any g i as in (2.6) (y Theorem 2.4); g i,h i are given y (2.8) (2.9), with α i1,β i1 not oth 0 (y Theorem 2.5). Conjecture 2.3 asserts that the Multiplicative Rule of Three holds for g i,h i given y (2.6) (2.7), with α i1,β i1 not oth 0. Prolem 2.9. Find the most general setting (i.e., the weakest restrictions on the expressions g i,h i ) for which the Multiplicative Rule of Three holds. Example The Multiplicative Rule of Three fails for g 4 = (1+xu 8 )(1+xu 7 ), h 4 = (1+yu 8 )(1+yu 7 ), (2.10) g 3 = (1+xu 6 )(1+xu 5 ), h 3 = (1+yu 6 )(1+yu 5 ), (2.11) g 2 = (1+xu 4 )(1+xu 3 ), h 2 = (1+yu 4 )(1+yu 3 ), (2.12) g 1 = (1+xu 2 )(1+xu 1 ), h 1 = (1+yu 2 )(1+yu 1 ). (2.13) In other words, the relations on 8 elements u 1,...,u 8 of a ring R resulting from the conditions g S h S = h S g S for all susets S of cardinality 3, with the g i and the h i given y (2.10) (2.13), do not imply the relation g S h S = h S g S for S = {1,2,3,4}. This was shown using a noncommutative Gröner asis calculation in Magma [8].

7 RULES OF THREE FOR COMMUTATION RELATIONS 7 Example For an 8-letter word z = z 1 z 2 z 3 z 4 z 1 z 2 z 3 z 4 in the alphaet {x,y}, let g i = 1+z i u i, h i = 1+z iv i. Inthissetting, themultiplicative RuleofThree (withn = 4)holdsfor222ofthe2 8 = 256 choices of z, and fails for the remaining 34. Specifically, the rule holds unless z 1 = z 1, z 2 = z 2, z 3 = z 3, z 4 = z 4, or z 1 < z 1, z 2 z 2, z 3 z 3, z 4 < z 4, or z 1 > z 1, z 2 z 2, z 3 z 3, z 4 > z 4, where we use the order relation x < y on the symols x and y. This was shown using a noncommutative Gröner asis calculation in Magma [8]. The Multiplicative Rule of Three always holds in the simplified version of Example 2.11 wherein the factors g i,h i depend on a single set of noncommuting variales: Theorem The Multiplicative Rule of Three holds when g i = 1+(α i x+α i y)u i and h i = 1+(β i x+β i y)u i for any α i,α i,β i,β i Q. (Here we use the notational conventions of Definition 2.8.) Theorem 2.12 is a special case of a more general result, see Theorem 5.3. Since the Multiplicative Rule of Three does not always hold, it is natural to consider Rules of Four and eyond (though this has not een the main focus of our investigation). For example, we can generalize Definition 2.8 as follows: for any k 0, we say that the Multiplicative Rule of k holds for g 1,...,g N,h 1,...,h N if for any quotient ring R = A/I, the following are equivalent: g S h S h S g S mod I for all susets S {1,...,N}; g S h S h S g S mod I for all susets S of cardinality k. Conjecture For any k 0 and N > k, the Multiplicative Rule of k fails for g i = 1+xu i, h i = 1+xv i, i = 1,2,...,N. Conjecture 2.13 would imply the failure of the group-theoretic Rule of k, for any k: Conjecture Fix integers k 0 and N > k, and consider the group whose presentation is given y generators g 1,...,g N,h 1,...,h N and relations g S h S = h S g S for S k. Then g S h S h S g S for any S > k. We verified Conjecture 2.13 (hence Conjecture 2.14) in the cases k 5 via a noncommutative Gröner asis calculation. 3. Rules of Three for sums and products Definition 3.1. Let M e a monoid. A suset M M is called potentially invertile if M can e emedded into a larger monoid in which all elements of M have left and right inverses (necessarily equal to each other); see [9, Section VII.3]. Similarly, a suset R of a ring R is potentially invertile if R can e emedded into a larger ring in which all elements of R are invertile.

8 8 JONAH BLASIAK AND SERGEY FOMIN By ause of terminology, we say that elements g 1,...,g N of a monoid (or ring) are potentially invertile if {g 1,...,g N } is a potentially invertile suset. Let R[x] e the ring of polynomials in one formal (central) variale x, with coefficients in a ring R. Then any suset of R[x] consisting of polynomials with constant term 1 is potentially invertile. Indeed, R[x] can e emedded into the ring R[[x]] of formal power series over R, and each polynomial with constant term 1 is invertile in R[[x]]. Throughout this paper, we use the notation [g,h] = gh hg for the commutator of elements g, h of an associative ring. Theorem 3.2 (The Rule of Three for sums vs. products). Let R e a ring, and let v 1,...,v N R andg 1,...,g N R, with g 1,...,g N potentiallyinvertile. Then the following are equivalent: [ i S v i,g S ] = 0 for all susets S {1,...,N}; [ i S v i,g S ] = 0 for all susets S of cardinality 3. Theorem 3.2 can e generalized to a setting of algeras with derivations. Recall that a derivation on a Q-algera R is a Q-linear map : R R satisfying Leiniz s law (fg) = (f)g +f (g). Theorem 3.3 (The Rule of Three for derivations). Let R e a Q-algera. Let 1,..., N e derivations on R, and let g 1,...,g N e potentially invertile elements of R satisfying a (g a ) = 0 for all 1 a N; (3.1) ( + a )(g g a ) = 0 for all 1 a < N; (3.2) ( c + + a )(g c g g a ) = 0 for all 1 a < < c N. (3.3) Then ( N + N )(g N g N 1 g 1 ) = 0. Theorem 3.4 (The Rule of Three for products vs. products and sums). Let R e a ring, and let g 1,...,g N,h 1,...,h N R e potentially invertile. Then the following are equivalent: [g S,h S ] = [g S, i S h i] = 0 for all susets S {1,...,N}; [g S,h S ] = [g S, i S h i] = 0 for all susets S of cardinality 3. Theorem 3.5. Let R e a ring, and let g 1,...,g N,h 1,...,h N R e potentially invertile elements satisfying the relations [h a,g a ] = 0 for all 1 a N; (3.4) [h +h a,g +g a ] = 0 for all 1 a < N; (3.5) [h +h a,g g a ] = 0 for all 1 a < N; (3.6) [h c +h +h a,g c g g a ] = 0 for all 1 a < < c N; (3.7) [g +g a,h h a ] = 0 for all 1 a < N; (3.8) [g c +g +g a,h c h h a ] = 0 for all 1 a < < c N. (3.9) Then [ i S g i, i S h ] [ i = i S g ] [ i,h S = gs, i S h i] =[gs,h S ]=0 for all susets S.

9 RULES OF THREE FOR COMMUTATION RELATIONS 9 Remark 3.6. It is easy to see that the relations (3.4) (3.5) alone imply [ i S g i, i S h i] = 0 for all S {1,...,N}; this implication can e regarded as an Additive Rule of Two. Theorem 3.5 implies (and so can e regarded as a strengthening of) the following rule. Corollary 3.7 (The Rule of Three for products and sums vs. products and sums). Let R e a ring, and let g 1,...,g N,h 1,...,h N R e potentially invertile. Then the following are equivalent: [ i S g i, i S h ] [ i = i S g ] [ i,h S = gs, i S h i] = [gs,h S ] = 0 for all S; [ i S g i, i S h ] [ i = i S g ] [ i,h S = gs, i S h i] = [gs,h S ] = 0 for all S 3. We note that Corollary 3.7 is also immediate from Theorem 3.4 and Remark 3.6. Remark 3.8. The cases S 3 of Theorem 3.5 hold without the requirement of potential invertiility; this can e verified y a noncommutative Gröner asis calculation. However, for S 4, this requirement cannot e dropped. More precisely, in the free associativealgeraq g 1,g 2,g 3,g 4,h 1,h 2,h 3,h 4,thetwo-sidedidealgeneratedytheleft-hand sides of (3.4) (3.9) does not contain the element [g 4 g 3 g 2 g 1,h 4 h 3 h 2 h 1 ] (nor does it contain [h 4 +h 3 +h 2 +h 1,g 4 g 3 g 2 g 1 ]). This was checked using a noncommutative Gröner asis calculation in Magma [8]. Remark 3.9. As explained in Section 7, Theorem 3.5 directly implies Theorem 1.1 via the sustitutions g i = 1+xu i, h i = 1+yv i. On the other hand, if we think of g i and h i in Theorem 3.5 asu i and v i, then Theorems 3.5 and1.1 are incomparale : in Theorem 3.5, we do not need the relations [h c +h +h a,g c g +g c g a +g g a ] = 0, (3.10) utwedorequireg i andh i toepotentiallyinvertile (cf.remark3.8), whiletheorem1.1 requires [e 1 (v S ),e 2 (u S )] = 0 for S = 3 ut not invertiility. To further clarify matters, we note that the assumptions of Theorem 3.5 (including invertiility) do not imply (3.10). To see this, take N =3, g i = 1+x 3 u 3 i +x4 u 4 i +x5 u 5 i, and h i = 1+yv i for i = 1,2,3. Impose relations on the u i,v i derived from the conditions g S h S = h S g S for all S of cardinality 3. Then relations (3.4) (3.9) hold ut (3.10) (with (a,, c) = (1, 2, 3)) does not. This was checked via a noncommutative Gröner asis computation. Theorem 3.2 and Theorems elow form a natural progression. Theorem 3.10 (The Rule of Three for products vs. sums and quadratic forms). Let R e a ring. Let v 1,...,v N R and g 1,...,g N R, with g 1,...,g N potentially invertile. Then the following are equivalent: g S e l (v S ) = e l (v S )g S for all susets S {1,...,N} and l 2; g S e l (v S ) = e l (v S )g S for all susets S of cardinality 3 and l 2.

10 10 JONAH BLASIAK AND SERGEY FOMIN Theorem 3.11 (The Rule of Three for products vs. elementary symmetric functions). Let R e a ring. Let v 1,...,v N R and g 1,...,g N R, with g 1,...,g N potentially invertile. Then the following are equivalent: g S e l (v S ) = e l (v S )g S for all susets S {1,...,N} and all l; g S e l (v S ) = e l (v S )g S for all susets S of cardinality 3 and all l. Remark Theorem 3.11 implies a variant of the Multiplicative Rule of Three (see Corollary 7.6) which generalizes Theorem 2.4 (which in turn generalizes Corollary 2.1). 4. Dehn diagrams. Group-theoretic lemmas For the purposes of this paper, a Dehn diagram (a simplified version of the notion of van Kampen diagram, see, e.g., [20, Section 4]) is a planar oriented graph whose edges are laeled y elements of a group, so that each cycle corresponds to a relation in the group. A more precise formulation is given in Definition 4.1 elow. Definition 4.1. Let D e a finite oriented graph properly emedded in the real plane; that is, it is drawn so that its edges only meet at common endpoints. We require each vertex of D to have at least two incident edges. The complement of D in the plane is a disjoint union of faces: some ounded faces homeomorphic to disks, and a single outer face. Assume that every edge of D has een laeled y an element of a group G. Such an edge-laeled oriented graph is called a Dehn diagram if the product along the oundary F of each ounded face F is equal to 1. More precisely, starting with an aritrary vertex on F and moving either clockwise or counterclockwise, we multiply the elements of G associated with the edges, inverting them when moving against the orientation of an edge. It is easy to see that this condition does not depend on the starting location on F. The following simple ut useful oservation goes ack to M. Dehn. Lemma 4.2. In a Dehn diagram, the product of laels along the oundary of the outer face is equal to 1. Below we present several group-theoretic results in the spirit of (multiplicative) Rules of Three, cf. Prolem 2.2. All the proofs utilize Dehn diagrams. Proposition 4.3. Let G e a group, and let a,,c,a,b,c G satisfy ac = Ca, B = B, ca = Ac, aba = BAa, ccb = CBc. (4.1) Then cacba = CBAca. Proof. In the Dehn diagram shown in Figure 1, each ounded face commutes. Hence so does the outer face, and the claim follows. Proposition 4.3 is a special case of the following result.

11 RULES OF THREE FOR COMMUTATION RELATIONS 11 a C C a B c B A C B a B c A A c Figure 1. The proof of Proposition 4.3. Theorem 4.4. Let G e a group, and let the elements g 1,...,g N,h 1,...,h N G satisfy Then g a h c = h c g a for all a,c {1,...,N} with a c 2; (4.2) g a h a = h a g a for all 1 < a < N; (4.3) g a+1 g a h a+1 h a = h a+1 h a g a+1 g a for all 1 a < N. (4.4) g N g N 1 g 1 h N h N 1 h 1 = h N h N 1 h 1 g N g N 1 g 1. (4.5) Proof. TheproofisadirectgeneralizationoftheaoveproofofProposition4.3. Itrelieson the Dehn diagram shown in Figure 2, which illustrates the case N = 6. The quadrilateral and octagonal faces of the diagram correspond to relations (4.3) and (4.4), respectively. The ounded faces at the ottom and the top can e tiled y rhomi corresponding to the relations (4.2); to avoid clutter, these tiles are not shown. The outer oundary corresponds to (4.5). Remark 4.5. The statement of Proposition 4.3 (or its generalization, Theorem 4.4) does not involve inverses. As such, it readily extends to potentially invertile elements a,,c,a,b,c in a monoid M. Note however that the assumption of potential invertiility cannot e dropped: in an aritrary monoid M, elements a,,c,a,b,c satisfying relations (4.1) do not have to satisfy cacba = CBAca. On the other hand, the latter identity is implied y (4.1) under the additional assumption that B is potentially invertile. (There is no need to require anything else.) Remark 4.6. A. I. Malcev [17, 18] gave an infinite list of conditions (more precisely, quasi-identities) that a monoid M must satisfy in order to e emeddale into a group. (Note that emeddaility of M into a group is equivalent to the set of all elements of M eing potentially invertile. Malcev s argument is reproduced in [9, Section VII.3]; see also [14] for an alternative perspective.) Apart from left and right cancellativity, the simplest of those conditions is the following (cf. Figure 3): p,q,r,s,p,q,r,s M ((PQ = pq & RQ = rq & RS = rs) PS = ps). (4.6)

12 12 JONAH BLASIAK AND SERGEY FOMIN g 1 h 6 g 2 h 5 g 3 h 4 g 4 h 3 g 5 h 6 g 4 h 5 g 3 h 4 g 2 h 3 g 1 h 2 g 6 h 5 g 5 h 4 g 4 h 3 g 3 h 2 g 2 h 1 h 6 g 5 h 5 g 4 h 4 g 3 h 3 g 2 h 2 g 1 h 5 g 6 h 4 g 5 h 3 g 4 h 2 g 3 h 1 g 2 h 4 g 3 h 3 g 4 h 2 g 5 h 1 g 6 Figure 2. Proof of Theorem 4.4. It turns out that Theorem 4.4 holds for any monoid satisfying condition (4.6). Curiously, neither cancellativity nor other Malcev s conditions are required. P S Q R q r p s Figure 3. Dehn diagram illustrating Malcev s condition (4.6). Theorem 4.7. Let G e a group, and let g 1,...,g N,h 1,...,h N G e such that for 1 < < N, we have: if z G commutes with h 1 g, then z commutes with oth g and h. (4.7) Then the following are equivalent: g S h S = h S g S for all susets S {1,...,N}; g S h S = h S g S for all susets S of cardinality 3. In Section 5, we show that condition (4.7) is satisfied in the setting of Theorem 2.5. This enales us to deduce the latter from Theorem 4.7.

13 RULES OF THREE FOR COMMUTATION RELATIONS 13 The proof of Theorem 4.7 relies on two group-theoretic lemmas. Lemma 4.8. Let G e a group, and let g a,g,g c,h a,h,h c G satisfy g a h a = h a g a, g h = h g, g c h c = h c g c, (4.8) g g a h h a = h h a g g a, g c g a h c h a = h c h a g c g a, g c g h c h = h c h g c g. (4.9) Then, if one of the following two relations holds in G, so does the other: h 1 g (gc 1 h a g c h 1 a ) = (gc 1 h a g c h 1 a )h 1 g ; (4.10) g c g g a h c h h a = h c h h a g c g g a. (4.11) Proof. It suffices to oserve that in the Dehn diagram in Figure 4, the outer face corresponds to the relation (4.11); the 12-gon in the middle corresponds to the relation (4.10); the other ounded faces correspond to relations (4.8) (4.9). g a h c g c h a g h c h c g a g a g c h a g c h a h h g g g c g h g c h a g c h a h h a h g h c g a Figure 4. The proof of Lemma 4.8. Lemma 4.9. Let G e a group, and let g 1,...,g N,h 1,...,h N G satisfy g a h a = h a g a for all 1 a N, (4.12) g g a h h a = h h a g g a for all 1 a < N, (4.13) Then g (g 1 c h a g c h 1 a ) = (g 1 c h a g c h 1 a )g for all 1 a < < c N, (4.14) h 1 (g 1 c h a g c h 1 a ) = (g 1 c h a g c h 1 a )h 1 for all 1 a < < c N. (4.15) g N g N 1 g 1 h N h N 1 h 1 = h N h N 1 h 1 g N g N 1 g 1. (4.16)

14 14 JONAH BLASIAK AND SERGEY FOMIN Proof. Induction on N. The cases N = 1 and N = 2 are immediate from (4.12) (4.13). The case N = 3 follows from Lemma 4.8 since the relations (4.14) (4.15) imply (4.10). Now assume N 4. By the inductive hypothesis, the following relations hold: g N 1 g N 2 g 2 h N 1 h N 2 h 2 = h N 1 h N 2 h 2 g N 1 g N 2 g 2. (4.17) g N 1 g N 2 g 1 h N 1 h N 2 h 1 = h N 1 h N 2 h 1 g N 1 g N 2 g 1. (4.18) g N g N 1 g 2 h N h N 1 h 2 = h N h N 1 h 2 g N g N 1 g 2. (4.19) It remains to verify that the relations (4.12) (4.15) and (4.17) (4.19) imply (4.16). The Dehn diagram in Figure 5 illustrates the argument in the case N = 4. In the diagram, the leftmost ounded face corresponds to the relation (4.19); the rightmost ounded face corresponds to the relation (4.18); the octagonal face on the left corresponds to the relation (4.17); the octagonal face at the top corresponds to the relation (4.13); the two quadrilateral faces correspond to the relation (4.12); the four inner rectangles correspond to the relations (4.14) (4.15); and the outer face corresponds to (4.16). The general case is similar, with N rectangles in the center. g 1 h 4 g 4 h 1 g 2 h 4 h 4 g 1 g 1 g 4 h 1 g 4 h 1 h 3 h 3 g 2 g 2 g 4 h 1 g 4 h 1 g 3 h 2 g 3 g 3 g 4 h 1 g 4 h 1 h 2 h 3 h 3 g 2 g 4 h 1 g 4 h 1 g 3 h 2 h 2 g 4 g 4 h 1 g 4 h 1 h 1 h 4 h 3 h 2 g 3 g 2 g 1 Figure 5. The proof of Lemma 4.9.

15 RULES OF THREE FOR COMMUTATION RELATIONS 15 Proof of Theorem 4.7. For 1 a < < c N, conditions of Theorem 4.7 include (4.8), (4.9), and (4.11). By Lemma 4.8, relation (4.10) follows. In view of condition (4.7), we then otain (4.14) (4.15). Hence Lemma 4.9 applies, yielding (4.16), as desired. We conclude this section y slightly strengthening Theorem 4.7, see Corollary 4.11 elow. This will require the following immediate consequence of Lemma 4.8. Lemma Let G e a group, and let g a,g,g c,h a,h,h c G satisfy g a h a = h a g a, g = h, g c h c = h c g c, g g a h h a = h h a g g a, g c g a h c h a = h c h a g c g a, g c g h c h = h c h g c g. Then g c g g a h c h h a = h c h h a g c g g a. Corollary Let G e a group. Let g 1,...,g N,h 1,...,h N G e such that for each 1 < < N, either g = h or condition (4.7) holds. Then the following are equivalent: g S h S = h S g S for all susets S {1,...,N}; g S h S = h S g S for all susets S of cardinality 3. Proof. Induction on N. For N 3, the result is clear. Now assume N 4. If g = h for some {2,...,N 1}, then the claim g N g 1 h N h 1 = h N h 1 g N g 1 followsy comining theinduction assumption withlemma 4.10, forg a = g 1 g 1, g c = g N g +1, h a = h 1 h 1, h c = h N h +1. In the only remaining case, condition (4.7) holds for all {2,...,N 1}, and the claim follows from Theorem Proofs of Theorems 2.5 and 2.12 We otain Theorem 2.5 y comining Theorem 4.7 with Lemma 5.2 elow. While Theorem 4.7 is purely group-theoretic, the proof of Lemma 5.2 implicitly relies on a Lagrange inversion argument for formal power series. As efore, we denote y R[[x,y]] the ring of formal power series in x and y, with coefficients in the (unital) Q-algera R. We use the notation R[[x,y]] for the multiplicative sugroup of R[[x,y]] formed y power series with constant term 1. Thus the elements of R[[x,y]] are formal expressions z = k=0 z k, with each z k R[x,y] a homogeneous polynomial of degree k in x and y, with coefficients in R. (From now on, we adopt the convention deg(x)=deg(y)=1.) We have z R[[x,y]] if and only if z 0 = 1 R. Lemma 5.1. Let q Q[[x,y]] and r R[[x,y]], with q 0 and r 0. Then qr 0. Proof. Define the lexicographic order on the monomials x i y j y setting x i y j x i y j def (i < i or (i = i and j < j )). The statement of the lemma is true for q Q and r R. Consequently the leading term of the power series qr, with respect to the lexicographic order, is the product of the leading terms of q and r, respectively. The lemma follows.

16 16 JONAH BLASIAK AND SERGEY FOMIN Lemma 5.2. Let ϕ 1,ϕ 2, Q[x,y] e homogeneous polynomials of degrees deg(ϕ k ) = k. Assume that ϕ 1 0. Letu R, andlet f = 1+ϕ 1 u+ϕ 2 u 2 + R[[x,y]]. If f commutes with z R[[x,y]], then u commutes with z. Proof. Let z = z 0 + z 1 + z 2 +, with z k R[x,y] a homogeneous polynomial of degree k. Then [f,z] = [f 1,z] = j 1 k 0 ϕ j[u j,z k ]. In order for this commutator to vanish, it must vanish in each degree. Since deg(ϕ j [u j,z k ]) = j + k, we conclude that 1 j m ϕ j[u j,z m j ] = 0 for every m 1. So we have: ϕ 1 [u,z 0 ] = 0 and ϕ 1 0, hence [u,z 0 ] = 0 (y Lemma 5.1); ϕ 1 [u,z 1 ]+ϕ 2 [u 2,z 0 ] = ϕ 1 [u,z 1 ] = 0, hence [u,z 1 ] = 0; ϕ 1 [u,z 2 ]+ϕ 2 [u 2,z 1 ]+ϕ 3 [u 3,z 0 ] = ϕ 1 [u,z 2 ] = 0, hence [u,z 2 ] = 0; and so on. We conclude that [u,z] = [u,z 0 +z 1 +z 2 + ] = 0, as desired. Proof of Theorem 2.5. Let us denote f = h 1 g = (1 β 1 yu + )(1+α 1 xu + ) = 1+(α 1 x β 1 y)u +. By Theorem 4.7, it suffices to verify that if f commutes with z R[[x,y]], then so do oth g and h. Indeed, we have α 1 x β 1 y 0, so Lemma 5.2 applies; hence u commutes with z, and therefore so do g and h. Replacing Theorem 4.7 y Corollary 4.11 in the aove argument, we otain a stronger version of the Multiplicative Rule of Three: Theorem 5.3. Let R e a Q-algera, and let u 1,...,u N R. Let g 1,...,g N,h 1,...,h N R[[x,y]] e of the form g i = 1+α i1 u i +α i2 u 2 i +α i3u 3 i +, (5.1) h i = 1+β i1 u i +β i2 u 2 i +β i3u 3 i +, (5.2) where α ik,β ik Q[x,y] are homogeneous polynomials of degree k. Assume that for every {2,...,N 1}, either g = h or α 1 β 1. Then the following are equivalent: g S h S = h S g S for all susets S {1,...,N}; g S h S = h S g S for all susets S of cardinality 2 and 3. Proof of Theorem It suffices to note that the assumptions in Theorem 5.3 are satisfiedwheng i = 1+α i1 u i andh i = 1+β i1 u i, foranylinearpolynomialsα i1,β i1 Q[x,y]. 6. Proof of Theorem 3.2 Lemma 6.1. Let R e a ring, and let g a,g,v a,v R satisfy [v a,g a ] = [v,g ] = 0, [v +v a,g g a ] = 0. Then [v a,g ]g a = g [g a,v ].

17 Proof. This follows from the identity RULES OF THREE FOR COMMUTATION RELATIONS 17 [v a,g ]g a g [g a,v ] = [v +v a,g g a ] g [v a,g a ] [v,g ]g a. Lemma 6.2. Let R e a ring, and let g a,g,g c,v a,v,v c R satisfy Then [v a,g c ]g g a = g c g [g a,v c ]. Proof. This follows from the identity [v a,g c ]g g a g c g [g a,v c ] [v,g ] = 0, [v +v a,g g a ] = [v c +v,g c g ] = 0, [v c +v +v a,g c g g a ] = 0. = [v c +v +v a,g c g g a ] g c [v +v a,g g a ] [v c +v,g c g ]g a +g c [v,g ]g a. Lemma 6.3. Let R e a monoid (or a ring), and let g 1,...,g m,z,z R satisfy If g 1 and g m are potentially invertile, then zg 1 = g m z, (6.1) zg g 1 = g m g z for all 1 < < m. (6.2) zg m 1 g m 2 g 1 = g m g m 1 g 2 z. (6.3) Proof. Passing to an extension of R wherein g 1 and g m have inverses, let us denote r = gm 1z = z g1 1 (cf. (6.1)). Condition (6.2) means that r commutes with g for 1<<m. Hence r commutes with g m 1 g 2, which is nothing ut (6.3). Corollary 6.4. Let R e a ring and let v 1,...,v N,g 1,...,g N R with g 1,...,g N potentially invertile. Suppose [ i S v i,g S ] = 0 for all S {1,...,N} of cardinality 3. Then for each suset S = {s 1 < < s m } {1,...,N}, [v s1,g sm ]g sm 1 g sm 2 g s1 = g sm g sm 1 g s2 [g s1,v sm ]. Proof. Apply Lemmas 6.1, 6.2, and 6.3, with z = [v s1,g sm ] and z = [g s1,v sm ]. Proof of Theorem 3.2. We need to show that relations [v a,g a ] = 0 for all 1 a N; (6.4) [v +v a,g g a ] = 0 for all 1 a < N; (6.5) [v c +v +v a,g c g g a ] = 0 for all 1 a < < c N (6.6)

18 18 JONAH BLASIAK AND SERGEY FOMIN imply [v sm +v sm 1 + +v s1,g S ] = 0 for any suset S={s 1 < <s m } {1,...,N}. We estalish this claim y induction on m = S. The cases m 3 are covered y (6.4) (6.6). Using the induction assumption and the Leiniz rule for commutators, we get: [v sm +v sm 1 + +v s1,g sm g s1 ] = [(v sm + +v s2 )+(v sm 1 + +v s1 ) (v sm 1 + +v s2 ),g sm g s1 ] = [v sm + +v s2,g sm g s2 ]g s1 +g sm g s2 [v sm + +v s2,g s1 ] +[v sm 1 + +v s1,g sm ]g sm 1 g s1 +g sm [v sm 1 + +v s1,g sm 1 g s1 ] [v sm 1 + +v s2,g sm ]g sm 1 g s1 g sm [v sm 1 + +v s2,g sm 1 g s2 ]g s1 g sm g s2 [v sm 1 + +v s2,g s1 ] = [v s1,g sm ]g sm 1 g s1 g sm g s2 [g s1,v sm ] = 0, where the last equality is y Corollary 6.4. Proof of Theorem 3.3. This theorem is proved y exactly the same argument as the one used for Theorem Proofs of Theorems 1.1, 3.4, 3.5, 3.10, and 3.11 Lemma 7.1. Let R e a ring, and let g a,g,g c,h a,h,h c R, with h potentiallyinvertile, satisfy the relations Then the following are equivalent: g h = h g, (7.1) g g a h h a = h h a g g a, (7.2) g c g h c h = h c h g c g. (7.3) g c g g a h c h h a = h c h h a g c g g a, g c g [g a,h c ]h h a = h c h [h a,g c ]g g a. Proof. The statement follows from the identity (in the appropriate extension of R): [g c g g a,h c h h a ] = g c g [g a,h c ]h h a h c h [h a,g c ]g g a (7.4) +[g c g,h c h ]h 1 g a h h a +h c h g c h 1 ([g g a,h h a ] [g,h ]h 1 g a h h a ). Lemma 7.2. Let R e a ring, and let g a,g,g c,h a,h,h c e potentially invertile elements of R satisfying (7.1) (7.3). Then any two of the following conditions imply the third: g c g g a h c h h a = h c h h a g c g g a, g c g [g a,h c ] = [h a,g c ]g g a, h c h [h a,g c ] = [h a,g c ]h h a.

19 Proof. Adding the trivial identity to (7.4), we otain RULES OF THREE FOR COMMUTATION RELATIONS 19 0 = [h a,g c ]g g a h h a +[h a,g c ]h h a g g a +[h a,g c ][g g a,h h a ] [g c g g a,h c h h a ] = (g c g [g a,h c ] [h a,g c ]g g a )h h a (h c h [h a,g c ] [h a,g c ]h h a )g g a +[h a,g c ][g g a,h h a ]+[g c g,h c h ]h 1 g a h h a +h c h g c h 1 ([g g a,h h a ] [g,h ]h 1 g a h h a ), and the claim follows. Lemma 7.3. Let R e a (unital) ring, and let g a,g,h a,h e potentially invertile elements of R satisfying g a h a = h a g a and g h = h g. Then any two of the following conditions imply the third: g g a h h a = h h a g g a, g [g a,h ] = [h a,g ]g a, h [h a,g ] = [h a,g ]h a. Proof. This is the g = h = 1 case of Lemma 7.2, with a suitale change of notation. Theorem 7.4 elow, although not a Rule of Three, is a powerful result which will e used to prove Theorems 3.4 and 3.5. Theorem 7.4. Let g 1,...,g N,h 1,...,h N e potentially invertile elements of a ring R satisfying the relations g a h a = h a g a for all 1 a N; (7.5) g [g a,h ] = [h a,g ]g a for all 1 a < N; (7.6) g c g [g a,h c ] = [h a,g c ]g g a for all 1 a < < c N; (7.7) h [h a,g ] = [h a,g ]h a for all 1 a < N; (7.8) h c h [h a,g c ] = [h a,g c ]h h a for all 1 a < < c N. (7.9) Then g S h S = h S g S for any S {1,...,N}. Proof. First, we claim that for any suset S={s 1 < < s m } {1,...,N}, one has [h s1,g sm ]g sm 1 g sm 2 g s1 = g sm g sm 1 g s2 [g s1,h sm ]. (7.10) This follows y applying Lemma 6.3 with z = [h s1,g sm ] and z = [g s1,h sm ]. (Conditions (6.1) (6.2) hold y (7.6) (7.7).) Again applying Lemma 6.3, this time with z = z = [h s1,g sm ] (and relying on (7.8) (7.9)), we get [h s1,g sm ]h sm 1 h sm 2 h s1 = h sm h sm 1 h s2 [h s1,g sm ]. (7.11) We now prove g S h S = h S g S y induction on m = S. The ase case m = 1 is given in (7.5). It remains to invoke Lemma 7.2 with making use of (7.10) and (7.11). g a = g s1, g = g sm 1 g s2, g c = g sm, h a = h s1, h = h sm 1 h s2, h c = h sm,

20 20 JONAH BLASIAK AND SERGEY FOMIN Proof of Theorem 3.4. We need to show that relations g a h a = h a g a for all 1 a N; (7.12) g g a h h a = h h a g g a for all 1 a < N; (7.13) g c g g a h c h h a = h c h h a g c g g a for all 1 a < < c N; (7.14) g g a (h +h a ) = (h +h a )g g a for all 1 a < N; (7.15) g c g g a (h c +h +h a ) = (h c +h +h a )g c g g a for all 1 a < < c N (7.16) imply g S h S = h S g S for all susets S. (The other conclusion is y Theorem 3.2.) By Theorem 7.4, the claim will follow once we have checked conditions (7.5) (7.9). Relation (7.12) is the same as (7.5). By Lemma 6.1, relations (7.12) and (7.15) imply (7.6). By Lemma 6.2, relations (7.12) and (7.15) (7.16) imply (7.7). Finally, y Lemmas , relations (7.6) (7.7) and (7.12) (7.14) imply (7.8) (7.9). Corollary 7.5. Let g 1,...,g N,h 1,...,h N e potentially invertile elements of a ring R satisfying [g a,h ] = [h a,g ] for all 1 a < N; (7.17) [g a,h a ] = 0 for all 1 a N; (7.18) g [g a,h ] = [h a,g ]g a for all 1 a < N; (7.19) g c g [g a,h c ] = [h a,g c ]g g a for all 1 a < < c N; (7.20) h [h a,g ] = [g a,h ]h a for all 1 a < N; (7.21) h c h [h a,g c ] = [g a,h c ]h h a for all 1 a < < c N. (7.22) Then g S h S = h S g S for any S {1,...,N}. Proof. Sustitute (7.17) into (7.21) (7.22) to get (7.8) (7.9); then apply Theorem 7.4. Proof of Theorem 3.5. We will use Corollary 7.5 to show that (3.4) (3.9) imply g S h S = h S g S for all susets S. (The other conclusions are y Theorem 3.2 and Remark 3.6.) Relations (7.17) (7.18) are equivalent to (3.4) (3.5). Relations (7.19) (7.20) (which are identical to (7.6) (7.7)) are checked precisely as in the proof of Theorem 3.4, using (3.6) (3.7) and Lemmas In the same way, we use(3.8) (3.9) to otain(7.21) (7.22). Proof of Theorem 1.1. Set g i = 1+xu i and h i = 1+yv i for i = 1,...,N. (Here, as efore, we are operating in the ring of formal power series in two variales x and y.) Then the g i and the h i are invertile. Furthermore, the relations (1.4) (1.8) imply the relations (3.4) (3.9). Applying Theorem 3.5, we conclude that g S h S = h S g S for any S {1,...,N}. Equivalently, e k (u S )e l (v S ) = e l (v S )e k (u S ) for all k, l, as desired. Proof of Theorem Seth i = 1+yf i R[y]; here, asefore, y isaformalvarialecommuting with all elements of R. One then checks that the two statements in Theorem 3.4 translate into the respective statements in Theorem The claim follows. Proof of Theorem Same argument as aove, this time with h i = 1+yf i R[y]/(y 3 ).

21 RULES OF THREE FOR COMMUTATION RELATIONS 21 We conclude this section with additional results on the Multiplicative Rule of Three, cf. Definition 2.8. Corollary 7.6. The Multiplicative Rule of Three holds for g i = 1 + α i1 x + α i2 x 2 + and h i = 1+yv i, for any α ik A = Q u 1,...,u M,v 1,...,v M. Proof. Let I e an ideal in A, and let I[[x,y]] e the ideal generated y I inside A[[x,y]]. Suppose [g S,h S ] 0 mod I[[x,y]] for all S 3. Taking the coefficient of y, we get [g S, i S v i] 0 mod I[[x,y]] andconsequently [g S, i S h i] 0 mod I[[x,y]]. It remains to apply Theorem 3.4 (with R = A[[x,y]]/I[[x,y]]). One can more generally identify specific conditions on the β ij in Conjecture 2.3 which ensure that g S ( i S h i) = ( i S h i)g S for all S 3. Here is one example. Corollary 7.7. The Multiplicative Rule of Three holds for g i = 1+α i1 x+α i2 x 2 + and h i = 1+β i1 yv i +β i4 y 4 v 4 i, for any α ik A and β i1,β i4 Q. Proof. Same argument as aove, this time noting that for d = 1,4 and S 3, taking the coefficient of y d in [g S,h S ] 0 yields [g S, i S β idvi d] Proof of Theorem 1.6 We will need the following slight generalization of Lemma 7.2. Lemma 8.1. Let R e a ring, and let g a,g,g c,h a,h,h c e potentially invertile elements of R satisfying (7.1) (7.3). Let z R. Then any two of the following conditions imply the third: g c g g a h c h h a = h c h h a g c g g a, g c g [g a,h c ] = zg g a, h c h [h a,g c ] = zh h a. Proof. This follows from a modified version of the identity in the proof of Lemma 7.2: [g c g g a,h c h h a ] = (g c g [g a,h c ] zg g a )h h a (h c h [h a,g c ] zh h a )g g a +z[g g a,h h a ]+[g c g,h c h ]h 1 g a h h a +h c h g c h 1 ([g g a,h h a ] [g,h ]h 1 g a h h a ). Lemma 8.2. Let A e a ring, and let u 1,...,u N,v 1,...,v N A satisfy [u a,v ] = [v a,u ] for all 1 a < N. (8.1) For every i {1,...,N}, let α i,β i e central elements of A such that 1+α i u i and 1+β i v i are invertile, and define g i,h i A y making one of the following two choices: { { g i = 1+α i u i, g i = (1+α i u i ) 1, either or (8.2) h i = 1+β i v i, h i = (1+β i v i ) 1.

22 22 JONAH BLASIAK AND SERGEY FOMIN Suppose that the following relations are satisfied: [v a,g a ] = 0 for all 1 a N; (8.3) [v +v a,g g a ] = 0 for all 1 a < N; (8.4) [v c +v +v a,g c g g a ] = 0 for all 1 a < < c N; (8.5) [u a,h a ] = 0 for all 1 a N; (8.6) [u +u a,h h a ] = 0 for all 1 a < N; (8.7) [u c +u +u a,h c h h a ] = 0 for all 1 a < < c N. (8.8) Then g N g N 1 g 1 h N h N 1 h 1 = h N h N 1 h 1 g N g N 1 g 1. Lemma 8.2 can e thought of as a fancy version of Theorem 3.5. Proof. Let P(m,n) e the statement of the lemma with 1 and N replaced y m and n, respectively. We will prove P(m,n) y induction on n m. The case n = m is the relation [g a,h a ] = 0 for all 1 a N, (8.9) which is immediate from (8.2) and (8.3) (or (8.6)). Now assume m < n. First notice that P(m,n) is equivalent to P(m,n) with gm+n i 1 in place of g i, h 1 m+n i in place of h i, and u m+n i (resp., v m+n i, α m+n i, β m+n i ) in place of u i (resp., v i, α i, β i ). For instance, [v c + v + v a,g c g g a ] = 0 is equivalent to [v c + v + v a,ga 1g 1 gc 1 ] = 0; the set of these relations over all m a < < c n is identical to the relations [v n+m c +v n+m +v n+m a,gn+m cg 1 1 n+m g 1 n+m a] = 0 over all m a < < c n. By the previous paragraph, we may assume that at least one of the following holds: g m = 1+α m u m and h m = 1+β m v m ; (8.10) g n = 1+α n u n and h n = 1+β n v n. (8.11) Let us assume (8.11), as the case of (8.10) is handled similarly. By (8.1) and (8.11), we have α n [u m,h n ] = α n β n [u m,v n ] = α n β n [v m,u n ] = β n [v m,g n ]. (8.12) By Corollary 6.4, the relations (8.3) (8.8) imply g n g n 1 g m+1 [g m,v n ] = [v m,g n ]g n 1 g n 2 g m, (8.13) h n h n 1 h m+1 [h m,u n ] = [u m,h n ]h n 1 h n 2 h m. (8.14) Next multiply (8.13), (8.14) y β n, α n, respectively, and apply (8.11) and (8.12) to otain g n g n 1 g m+1 [g m,h n ] = β n [v m,g n ]g n 1 g n 2 g m, (8.15) h n h n 1 h m+1 [h m,g n ] = β n [v m,g n ]h n 1 h n 2 h m. (8.16) Now P(m,n) follows y applying Lemma 8.1 with g a = g m, g = g n 1 g m+1, g c = g n, h a = h m, h = h n 1 h m+1, h c = h n, z = β n [v m,g n ], making use of (8.15) (8.16) and the induction assumption.

23 RULES OF THREE FOR COMMUTATION RELATIONS 23 Proof of Theorem 1.6. The statement isproved y applying Lemma 8.2with A = R[[x,y]] and g i,h i given y { { 1+xu i if i is unarred 1+yv i if i is unarred g i = h (1 xu i ) 1 i = (8.17) if i is arred, (1 yv i ) 1 if i is arred. (Thus α i = x,β i = y if i is unarred and α i = x,β i = y if i is arred.) We verify the relation (8.1) using the S 2, k = 1 cases of (1.18): [u a,v ] [v a,u ] = [u +u a,v +v a ] [u a,v a ] [u,v ] = 0. The remaining relations (8.3) (8.8) follow from (1.18) (1.19), using the fact that g S = k xk e k (u S ), h S = l yl e l (v S ), for any S {1,...,N}. So Lemma 8.2 applies, and Theorem 1.6 follows. Acknowledgments. We thank the participants of the conference Lie and Jordan Algeras VI (Bento Gonçalves, Decemer 2015) for stimulating discussions. We thank Elaine So for help typing and typesetting figures and the anonymous referee for a numer of helpful comments. References [1] S. H. Assaf, Dual equivalence graphs and a cominatorial proof of LLT and Macdonald positivity, arxiv: v5, Octoer [2] C. Benedetti and N. Bergeron, Fomin-Greene monoids and Pieri operations, Algeraic monoids, group emeddings, and algeraic cominatorics, Fields Inst. Commun. 71, , Springer, [3] J. Blasiak, Kronecker coefficients for one hook shape, arxiv: [4] J. Blasiak, What makes a D 0 graph Schur positive? J. Algeraic Comin. (2016), [5] J. Blasiak, Haglund s conjecture on 3-column Macdonald polynomials, Math. Z. 283 (2016), [6] J. Blasiak and S. Fomin, Noncommutative Schur functions, switchoards, and Schur positivity, arxiv: , to appear in Selecta Math. [7] J. Blasiak and R. I. Liu, Kronecker coefficients and noncommutative super Schur functions, arxiv: [8] W. Bosma, J. Cannon, and C. Playoust, The Magma algera system. I. The user language, J. Symolic Comput. 24 (1997), [9] P. M. Cohn, Universal algera, Dordrecht, [10] S. Fomin and C. Greene, Noncommutative Schur functions and their applications, Discrete Math. 193 (1998), [11] A. N. Kirillov, Notes on Schuert, Grothendieck and key polynomials, SIGMA 12 (2016), 57 pp. [12] T. Lam, Affine Stanley symmetric functions, Amer. J. Math. 128 (2006), [13] T. Lam, Rion Schur operators, European J. Comin. 29 (2008), [14] J. Lamek, The immersiility of a semigroup into a group, Canadian J. Math. 3 (1951), [15] A. Lascoux and M.-P. Schützenerger, Le monoïde plaxique, Noncommutative structures in algera and geometric cominatorics (Naples, 1978), , Quad. Ricerca Sci., 109, CNR, [16] R. I. Liu, A simplified Kronecker rule for one hook shape. arxiv: [17] A. Malcev, On emedding of associative systems into groups [in Russian], Mat. Sornik N.S. 6 (48) (1939),

24 24 JONAH BLASIAK AND SERGEY FOMIN [18] A. Malcev, On emedding of associative systems into groups. II [in Russian], Mat. Sornik N.S. 8 (50) (1940), [19] J.-C. Novelli and A. Schilling, The forgotten monoid, Cominatorial representation theory and related topics, 71 83, RIMS Kôkyûroku Bessatsu, B8, Res. Inst. Math. Sci. (RIMS), Kyoto, [20] D. Peifer, Max Dehn and the origins of topology and infinite group theory, Amer. Math. Monthly 122 (2015), [21] M.-P. Schützenerger, La correspondance de Roinson, Cominatoire et représentation du groupe symétrique, , Lecture Notes in Math., 579, Springer, Berlin, [22] R. P. Stanley, Enumerative cominatorics, vol. 2, Camridge University Press, Camridge, Department of Mathematics, Drexel University, Philadelphia, PA address: jlasiak@gmail.com Department of Mathematics, University of Michigan, Ann Aror, MI address: fomin@umich.edu