Unified characterizations of minuscule Kac-Moody representations built from colored posets

R.M. Green described structural properties that"doubly infinite"colored posets should possess so that they can be used to construct representations of most affine Kac-Moody algebras. These representations are analogs of the minuscule representations of the semisimple Lie algebras, and his posets ("full heaps") are analogs of the finite minuscule posets. Here only simply laced Kac-Moody algebras are considered. Working with their derived subalgebras, we provide a converse to Green's theorem. Smaller collections of colored structural properties are also shown to be necessary and sufficient for such poset-built representations to be produced for smaller subalgebras, especially the"Borel derived"subalgebra. The representations of the Borel derived subalgebras are the restrictions of the $\lambda$-minuscule Demazure modules when the poset is finite. These developments lead to the formulation of unified definitions of finite and infinite colored minuscule and $d$-complete posets. This paper launches a program that seeks to extend the notion of"minuscule representation"to Kac-Moody algebras, and to classify such representations. Knowledge of Lie theory is not needed to read the core content of this paper; it is needed only for the motivations.


Introduction
Any mathematician can read the core content of this paper, since it consists of constructing linear operators on vector spaces defined with partially ordered sets and then computing commutator brackets of operators.
While knowledge of representation theory is needed for the motivations, no Lie theory is employed.
For most affine Kac-Moody algebras, R. M. Green constructed [Gr1] a small number of beautiful representations whose weight diagrams were unbounded above and below; we refer to such structures as being "doubly infinite." In contrast, the familiar Category O representations have weight diagrams that are bounded above. Green's representations and the doubly infinite posets from which they were built formed a central topic in his 2013 Cambridge tract [Gr3]. He noted that these representations (with no highest weights) are More generally, Stembridge characterized the heaps for all λ-minuscule elements. M. Hagiwara described [Ha1,Ha2] the minuscule heaps for elements of the Kac-Moody Weyl groups specified by star shaped Dynkin diagrams and for the affine Weyl groupÃ n . Stembridge's coloring axioms for the d-complete posets are not all self-dual. Proctor showed that d-complete posets have unique jeu de taquin rectifications [Pr4] and (with D. Peterson's help) the hook length property [Pr5]. These posets have been receiving increasing attention, as in [KlRa]. There is a bibliography for them in [PrSc]. When that study of the axioms for finite uncolored d-complete posets was written, it became apparent that the definition of "d-complete" could likely be extended to infinite locally finite posets. But it was unclear precisely what the most appropriate definition should be for such posets. For further historical details, see Section 13 of [Pr5].
Adopting some of Stembridge's axioms, Green axiomatically defined [Gr1] "full heaps" colored by Dynkin diagrams Γ. These are doubly infinite locally finite colored posets P in which the appearances of each color from Γ are unbounded above and below. He regarded these posets as being close companions to the finite minuscule posets. All of his coloring axioms for full heaps were self-dual. The "extended slant lattices" used by Hagiwara to describe the minuscule heaps for typeÃ n were early appearances of full heaps.
Let g ′ be the derived Kac-Moody algebra for a Dynkin diagram Γ. Green showed in Theorem 4.1.6(i) of [Gr3] that the lattice of splits of a full heap for Γ "carried" a representation of g ′ ; this had first appeared as Theorem 3.1 of [Gr1]. To save space, throughout this paper we restrict our attention to simply laced Γ.
For such diagrams Γ, our foremost new main result (the "necessary" direction of Theorem 11.2) provides a converse to Theorem 4.1.6(i) that includes finite dimensional representations as well as infinite dimensional representations. To state this converse, we formulate a notion of "P-minuscule" representation (Section 4): This is a representation of g ′ carried by F I(P) that "looks like" a minuscule representation of a semisimple Lie algebra. At the same time for simply laced Γ we obtain a version of the "sufficient" Theorem 4.1.6(i) of [Gr3] that now includes posets of unknown (finite or "mixed") cardinality. While doing this we produce several intermediate results: As more and more coloring properties are assumed for the poset P, the representations constructed have stronger and stronger algebraic properties. Most often these algebraic properties are the satisfaction of some of the defining relations for g ′ . Each of these collections of coloring properties is necessary as well as being sufficient for the collection of algebraic properties at hand. This development clarifies which collections of the coloring properties assumed in Theorem 4.1.6(i) correspond to which algebraic aspects of the representations. It also facilitates comparison with the collections of coloring properties considered by Stembridge in his parallel study of the reduced decompositions of a λ-minuscule Kac-Moody Weyl group element w.
By omitting a "down-only" coloring property required in Theorem 11.2, earlier in Theorem 10.1 we obtain a similar characterization of colored poset constructions of representations of just the "Borel derived" (Section 3) subalgebra b ′ + . When P is finite, these representations arise as the restrictions to b ′ + of the Demazure b + -modules for the dominant λ-minuscule w. (Even when P is finite, the Kac-Moody algebra g at hand can be infinite dimensional.) For this "up only" analog to Theorem 11.2, we introduce two new definitions. We formulate the notion of "upper P-minuscule" representation of b ′ + (Section 4). For the fourth corner of the conceptual 2 × 2 table, we formulate a notion of colored d-complete that works for infinite locally finite colored posets. So this desire to obtain a theorem for b ′ + analogous to Theorem 11.2 led to a precise definition for infinite colored d-complete posets.
After introducing full heaps, Green defined [Gr3] the notion of "principal subheap" for full heaps colored by affine Γ. These are finite colored posets. He showed that the principal subheaps of such a full heap are isomorphic to each other. Then he proved that the possible principal subheaps were exactly the preexisting finite colored minuscule posets. With Proctor in [PrSt] we continue to restrict to the simply laced case. There we show that Green's full heaps are exactly our infinite Γ-colored minuscule posets and that Green's principal subheaps (the pre-existing finite colored minuscule posets) are exactly our finite Γ-colored minuscule posets. The relationship between the finite Γ-colored minuscule posets and the infinite Γ-colored minuscule posets is entirely different here than in [Gr3].
To give an overview of this paper, for each simply laced Γ we visualize a 2 × 2 table of representation characterization problems. (Extensions of our results that also handle the non-simply laced case will be presented in [Str].) The rows of the table respectively pose existence problems for "P-minuscule" representations of g ′ and "upper P-minuscule" representations of b ′ + . The posets P that are shown to solve these problems are respectively the Γ-colored minuscule posets and the Γ-colored d-complete posets. The two columns of the table pertain to the cardinality of the poset P. Earlier work [Pr3,Ste] has either been restricted to finite posets P or has handled [Gr3] finite and infinite posets separately. Using a posteriori knowledge of the cardinality of P, the columns of the table indicate finite P or infinite P: In this paper we do not assume a priori (not even in the proofs) that the poset P at hand is known to be finite or infinite.
Our answers to these four existence problems are summarized in Theorem 12.1. Stembridge's dominant minuscule heaps (i.e. Proctor's colored d-complete posets) inhabit the lower left corner of our 2 × 2 table and Green's full heaps inhabit the upper right corner. The infinite Γ-colored d-complete posets that inhabit the lower right corner are new. The original colored minuscule posets inhabit the upper left corner.
It is the introduction of "frontier census" coloring properties for the poset P that enables us to provide definitions of "Γ-colored minuscule" and "Γ-colored d-complete" that are uniform across the 2 × 2 table of scenarios. Given an element y ∈ P that is an extreme element of P of color b ∈ Γ, these properties limit the number of elements that lie beyond y in P that have colors that are adjacent to b in Γ.
As is true for the finite minuscule case, when P is a full heap colored by Γ of untwisted affine type the structure F I(P) can be viewed as a crystal. Then F I(P) can be used to give a representation of the corresponding quantum affine algebra, as is described in Section 8 of Green's paper [Gr1]. Green also used F I(P) to construct representations of most of the affine Weyl groups in [Gr2].
Proctor classified [Pr2,Pr3] the finite colored d-complete posets for simply laced Γ. Stembridge extended [Ste] this classification to non-simply laced Γ. Green [Gr3] and McGregor-Dorsey [McG] classified the full heaps. In [PrSt] we will classify the infinite colored d-complete posets for simply laced Γ and list all of the posets that are organized by the conceptual 2 × 2 table above. Having the new "necessary" direction of Theorem 11.2 available will enable [PrSt] us to also classify the P-minuscule representations. This will be done by using that direction to combine two of the main results of [Gr3] and [McG], namely Theorem 4.1.6(i) and their classification of full heaps. This classification of P-minuscule representations will be a step in our minuscule Kac-Moody program: In an anticipated third paper we plan to present a definition of "abstract minuscule" representation for a Kac-Moody algebra. This definition will not refer to a poset that has been supplied a priori. Given such a representation, we believe we will be able to construct a poset P so that the given representation can be viewed as a P-minuscule representation.
After definitions are given in Sections 2-4, this paper has four parts. The first part, Sections 5 and 6, concerns representations of the Borel derived subalgebra b ′ + that are carried by the lattice of splits F I(P). Theorem 5.4 states that the possession of three of our earliest coloring properties by the poset P is equivalent to the existence of a representation of the smaller subalgebra n + ⊂ b ′ + that is carried by F I(P). Section 6 studies the extension of this representation of n + to the Borel derived subalgebra b ′ + by specifying the actions of the simple coroots (which form a basis of the Cartan derived subalgebra h ′ ). There is some freedom available for such an extension; the weight functions we introduce are accounting tools to keep track of the coroot actions. In the second part, Sections 7-9, we introduce a particular nice weight function. The prototypical minuscule representations of semisimple Lie algebras have weights along their "sl 2 strings" that are composed of eigenvalues from {−1, 0, +1} for the simple coroot actions. Our preferred weight function is defined in Section 7. In Proposition 8.1 we begin to obtain simple coroot actions with {−1, 0, +1} values when three coloring properties for P are present. We introduce the frontier census coloring properties in Section 9. Sections 10-12 form our third part. After obtaining our main results in Sections 10 and 11, we summarize them in Section 12 by presenting our new definitions of Γ-colored d-complete and Γ-colored minuscule posets. The fourth part, Sections 13 and 14, previews future work. In Section 13 we describe the classifications of our posets and compare them to the posets used by Proctor, Stembridge, and Green. In Section 14 we give algebraic comments and propose an abstract definition of "minuscule" representation of a Kac-Moody algebra.

Combinatorial definitions
Fix a partially ordered set P throughout. Letters such as z, y, x, . . . are used to denote elements of P. Consult [Sta] for the following terminology: comparable elements, covering relations and the Hasse diagram, antichains, closed and open intervals, connected posets, and direct sums of posets. We assume each P is locally finite; this means that all of its closed intervals are finite. We write x → y to indicate that x is covered by y. We say x and y are neighbors in P if x → y or y → x. A subset F ⊆ P is a filter of P if whenever x ∈ F and y ≥ x, we also have y ∈ F. Dually, a subset I ⊆ P is an ideal of P if whenever x ∈ I and y ≤ x, we also have y ∈ I. For each filter F of P there is a corresponding ideal I := P − F. Let F I(P) be the set of  all ordered pairs (F, I) such that F is a filter of P and I is its corresponding ideal: These are the splits of P.
The set F I(P) becomes a distributive lattice when it is ordered by inclusion of the ideals within the splits.  We equip P with a surjective coloring function κ : P → Γ. We say that P is a Γ-colored poset. See Various poset coloring properties will be precisely defined as needed; Table 2.1 indexes these forthcoming definitions. The poset displayed in Figure 2.1 satisfies all of these properties; the poset displayed in

Algebraic definitions
We regard the graph Γ as being a simply laced Dynkin diagram: Once Γ has been given a total ordering, the associated generalized Cartan matrix is [θ ab ] a,b∈Γ . The Kac-Moody algebra g with Cartan subalgebra h and subalgebras n + and n − are defined in [Kac] for a given Dynkin diagram Γ. The positive and negative Borel subalgebras are respectively b ± := h + n ± . We refer to the subalgebras of the derived subalgebra g ′ := [g, g] formed by intersections of h and b ± with g ′ as the Cartan derived subalgebra h ′ and the Borel derived subalgebras b ′ ± . The algebras n ± and the derived algebras are generated [Kac,Section 9.11] by subsets of the symbols {x a , y a , h a } a∈Γ subject to some defining relations. When Γ is simply laced, the relations are certain unions (specified below) of the following sets: The relations HX and HY can be condensed to [h b , x a ] = θ ab x a and [h b , y a ] = −θ ab y a for a, b ∈ Γ. The algebra h ′ is the Lie algebra generated by {h a } a∈Γ subject to the relation HH, and so it is abelian. The algebra n + (respectively n − ) is the Lie algebra generated by {x a } a∈Γ (respectively {y a } a∈Γ ) subject to the relations XX (respectively YY). The algebra b ′ + (respectively b ′ − ) is the Lie algebra generated by {x a , h a } a∈Γ (respectively {y a , h a } a∈Γ ) subject to the relations XX, HH, and HX (respectively YY, HH, and HY). We note The algebra g ′ is the Lie algebra generated by {x a , y a , h a } a∈Γ subject to all of the relations above. When g is finite dimensional (and consequently semisimple), the algebras h, b ± , and g are equal to their derived counterparts. Outside of Section 14, we are concerned only with the derived subalgebras and n ± .
Let V be any vector space. We say an operator T : V → V is square nilpotent if T 2 = 0. Consider actions on V of the generators x a , y a , and h a that are respectively given by operators X a , Y a , and H a in End(V) for all a ∈ Γ. If the X a (respectively Y a ) are square nilpotent for all a ∈ Γ, we say their actions are collectively X-square (respectively Y-square) nilpotent. An h ′ -weight basis of V is a basis B of V that simultaneously diagonalizes the operators {H a } a∈Γ . A weight function on B is a Γ-set of C-valued functions on B. Here the h ′ -weight of {H a } a∈Γ is the weight function {ξ a } a∈Γ satisfying H a .v = ξ a (v).v for every a ∈ Γ and every v ∈ B. The eigenvalue set for B is We recall a basic algebraic fact: Let k, l ≥ 1 and let L be the Lie algebra defined by generators g 1 , . . . , g k subject to relations r 1 , . . . , r l in the Lie bracket [·, ·] in L. Let G 1 , . . . , G k be elements of End(V). Let g i → G i for 1 ≤ i ≤ k be a bijection. Suppose under this bijection the operators G 1 , . . . , G k also satisfy the relations r 1 , . . . , r l , now in the commutator [·, ·] in End(V). Then the bijection induces a Lie algebra homomorphism from L to the subalgebra of gl(V) generated by G 1 , . . . , G k . Hence V is a representation of L. Our usage of the terminology above and this algebraic fact will leave the actions of x a , y a , and h a implicit, and so we will refer only to the operators X a , Y a , and H a in End(V).

Representations of Lie algebras built from colored posets
We return to our standard context that is established by the fixed locally finite poset P colored with the fixed finite simple graph Γ by κ, and having lattice of splits F I(P). All of our results are stated in this context.
Let V be the free complex vector space on F I(P). For each split (F, I), denote the corresponding vector in V by F, I . Every mention of diagonal operators is made with respect to the basis { F, I } (F,I)∈F I(P) for V. Let a ∈ Γ and let (F, I) be a split in F I(P). To guarantee the sums in the following two definitions are finite, temporarily assume all antichains in P are finite. Define the color raising operator X a by X a . F, I := F − x, I + x ; here the sum is taken over all elements x of color a that are minimal in F. Dually, define the color lowering operator Y a by Y a . F, I := F + y, I − y ; here the sum is taken over all elements y of color a maximal in I. Whenever we create such operators, it will be clear that these sums are finite. Linearly extend X a and Y a to all of V. For a ∈ Γ, the action of X a (respectively Y a ) on a basis vector F, I can be viewed as summing over all ways to move up (respectively down) in F I(P) from (F, I) by an edge colored a. We extend our standard context to include V (usually implicitly) and the {X a , Y a } a∈Γ .
We say F I(P) carries a representation of n + (respectively n − ) if the {X a } a∈Γ (respectively {Y a } a∈Γ ) satisfy XX (respectively YY) with respect to the commutator [·, ·] in End(V). We say F I(P) carries a representa- We say a representation of b ′ at other splits can be computed by working up through F I(P) using the relation HX. So one can see We say a representation of g ′ carried by It can be seen that P-minuscule representations of g ′ are X-and Y-square nilpotent.
One can confirm that the lattice F I(P) of splits displayed in Figure 2.1 carries a P-minuscule representation of g ′ .
5 Square nilpotent representations of n + and n − We establish our earliest equivalences between sets of coloring properties and sets of algebraic conditions.
Theorem 5.4 summarizes this section by listing three coloring properties for P that are necessary and sufficient for the specified {X a } a∈Γ actions to generate an X-square nilpotent representation of n + that is carried by F I(P).
Proposition 5.1. The following are equivalent: (i) The color raising operators {X a } a∈Γ are X-square nilpotent.
(ii) The following two properties are satisfied by P: For all a ∈ Γ and every split (F, I), the property EC implies that the sums defining X a . F, I and Y a . F, I are either single terms or are zero. From now on when we are creating these operators we will be assuming property EC holds. The properties EC and ND together ensure that no two edges of the same color are incident in the Hasse diagram of F I(P).
Proof. To prove (i) implies (ii), first suppose that EC fails. Then there is a color a ∈ Γ and incomparable elements x, y ∈ P a . Let F be the filter generated by x and y. Since 2 F − x − y, I + x + y is a term in the expansion of X 2 a . F, I , we have X 2 a . F, I 0. Now suppose ND fails. Then there is a color a ∈ Γ and neighbors x → y, with x, y ∈ P a . Let F be the filter generated by x. Since F − x − y, I + x + y is a term in the expansion of X 2 a . F, I , we have X 2 a . F, I 0.
Now suppose (ii) holds and fix a ∈ Γ. Let (F, I) ∈ F I(P). Suppose X a . F, I = F − x, I + x for some element x ∈ P a . Let y ∈ P a ; here y is comparable to x by EC. However, note y cannot cover x by ND. Hence y is not minimal in F − x. Since y ∈ P a was arbitrary, we have X a . F − x, I + x = 0. So for every color a ∈ Γ and (F, I) ∈ F I(P) we have X 2 a . F, I = 0. Dualize to obtain the equivalence of (ii) and (iii).
We get our first defining relation by strengthening ND: Lemma 5.2. Suppose P satisfies EC and ND. Then the following are equivalent: (ii) The following additional property is satisfied by P: (NA): Neighbors have adjacent colors.
Proof. To prove (i) implies (ii), first suppose (ii) fails. Then there exist neighbors x → y such that either κ(x) = κ(y) or κ(x) κ(y). By ND we have κ(x) κ(y), so κ(x) κ(y). Let a := κ(x) and b := κ(y). Let F be the filter generated by x. Since F − x − y, I + x + y is a term in the expansion of X b X a . F, I , we have X b X a . F, I 0. However, X a X b . F, I = 0 since the only minimal element in F has color a. Hence . F, I 0, and so (i) fails. Now suppose (ii) holds. Let a and b be distant colors. Let (F, I) be any split and without loss of generality assume X b X a . F, I 0. Then there are elements x and y such that κ(x) = a and κ(y) = b and By NA we know x and y are not neighbors. Thus they are incomparable Dualize to obtain the equivalence of (ii) and (iii).
We get our second defining relation by introducing a special case of the future key property I2A: Lemma 5.3. Suppose P satisfies EC and ND. Then the following are equivalent: (ii) The following additional property is satisfied by P: (I3ND): If three successive neighbors x → y → z form an interval in P, then x and z have different colors.
By Proposition 5.1 the first and last terms vanish when acting on any split. Thus the relation [X a , [X a , X b ]] = 0 holds if and only if X a X b X a = 0 holds. Suppose (i) holds, so that for all a, b ∈ Γ we have X a X b X a = 0. Suppose three successive neighbors x → y → z form an interval in P. Define a := κ(x) and b := κ(y) and c := κ(z). Let F be the filter generated Hence we have c a, so I3ND holds. Now suppose (ii) holds. Let a, b ∈ Γ and (F, I) ∈ F I(P) be such that X b X a . F, I 0. Then there are elements x and y with κ(x) = a and κ(y) = b such that X b X a . F, I = F − x − y, I + x + y . Either x and y are incomparable or x → y. Suppose x and y are incomparable. Every element of color a in F − x − y must be greater than x by EC, but none may cover x by ND. Hence X a X b X a . F, I = 0. Now suppose x → y.
Suppose z ∈ F − x − y has color a. By I3ND the set of elements {x, y, z} is not an interval. Thus z is not minimal in F − x − y, and so again X a X b X a . F, I = 0. Dualize to obtain the equivalence of (ii) and (iii).
Since NA implies ND, we can combine the three results above to produce: Theorem 5.4. The following are equivalent: (i) The lattice F I(P) carries an X-square nilpotent representation of n + .
(ii) The properties EC, NA, and I3ND are satisfied by P.
(iii) The lattice F I(P) carries a Y-square nilpotent representation of n − .
We remark that if a, b ∈ Γ are adjacent, then [X b , X a ] 0 if and only if there exist neighbors in P with those two colors. This will not be used here; see [Str] for details.
6 Square nilpotent representations of b ′ + and b ′ − We extend our representations of n + (respectively n − ) to b ′ + (respectively b ′ − ) by constructing diagonal operators {H a } a∈Γ on V that satisfy HX and HY. Locally, if two basis vectors are connected by an edge in F I(P), the relations HX (or HY) indicate how to relate the h ′ -weights of the two vectors: (a) The operators {X a , H a } a∈Γ satisfy the relations HX if and only if for every b ∈ Γ, split (F, I) ∈ F I(P), F, I , and so the equivalence in (a) follows. Dualize to obtain (c). Part (b) holds since any edge in F I(P) can be dually viewed both as transferring a minimal element of a filter to an ideal and as transferring a maximal element of an ideal to a filter.
We will need to compare the h ′ -weights of two splits at a distance. Partition F I(P) into components as This is the signed net number of edges of color b traversed in any finite path from (F, I) to (F ′ , I ′ ). Note that We now present a components-wide version of the equations in Lemma 6.1. Let {η a } a∈Γ be a weight function on F I(P). We call it a component weight function if for all b ∈ Γ, whenever (F, I) and (F ′ , I ′ ) are in the same component of F I(P) we have The following easy fact is confirmed [Str] by concatenating a path   Proof. Using Lemma 6.7(a), let {η a } a∈Γ be any component weight function. Then (iii) implies (ii) by Corollary 6.6(b) using {η a } a∈Γ . Also (ii) implies (i) by restricting to the operators {X a } a∈Γ . And (i) implies (iii) by Theorem 5.4. Dualize to obtain the equivalence of (iii), (iv), and (v). The last statement follows from Corollary 6.6(a).

A combinatorially motivated component weight function
We continue to assume P satisfies EC. If one does not care about the relationship between combinatorial properties and eigenvalues, Theorem 6.8 said that a representation of n + carried by F I(P) can be extended to b ′ + without requiring coloring properties for P beyond EC, NA, and I3ND: Create a component weight function by choosing for each component of F I(P) a Γ-set of complex numbers and a split. Then use the corresponding component diagonal operators {H a } a∈Γ to extend the action of n + . Here we construct a particular weight function {µ a } a∈Γ on F I(P) whose values are determined by the local structure of P.
When P has two new additional properties beyond EC, in Proposition 7.1 we use Corollary 6.4 to show that it is a component weight function. As we work toward obtaining the attractive (upper) P-minuscule representations of g ′ (and b ′ + ), in the next two sections we will obtain relationships between further coloring properties and this h ′ -weight.
We prepare to define our new Z-valued weight function {µ a } a∈Γ . Fix a color b ∈ Γ. To construct  We also define ψ b : F I(P) → N in dually analogous stages. Let (F, I) be a split. If P b ∩ F does not have a minimal element, then set ψ b (F, I) := 1. Now suppose that P b ∩ F has a minimal element y. By EC the element y is unique. We build up another set Ψ b (F, I) ⊆ F from the empty set ∅. Let z ∈ F. We place z into Ψ b (F, I) if it meets the following three requirements: (i) The element z is less than y, (ii) Its color c := κ(z) is adjacent to b, and (iii) The number of elements less than y that are in P c ∩ F is finite.
If there is some color a ∼ b such that there are infinitely many elements less than y in P a ∩ F, then set It is easy to see that this new property I2A implies both ND and I3ND.
Proof. We use Corollary 6.4 to show {µ a } a∈Γ is a component weight function. Let b ∈ Γ, let (F, I) ∈ F I(P), and let x be minimal in F. Define a := κ(x). We must show µ b (F − x, I + x) − µ b (F, I) = θ ab .
First suppose a = b. Start with the case P a ∩ I = ∅. Since x is minimal in F, we have ψ a (F, I) = |Ψ a (F, I)| = 0 and µ a (F, I) = −1. Here P a ∩(I + x) ∅. Since x is maximal in I + x, we have υ a (F − x, I + x) = |Υ a (F − x, I + x)| = 0 and µ a (F − x, I + x) = 1. We get µ a (F − x, I + x) − µ a (F, I) = 2 = θ aa . Otherwise we have the case P a ∩ I ∅. Let z ∈ P a ∩ I and note that P a ∩ [z, x] is finite by local finiteness for [z, x].
So P a ∩ I has a maximal element y. Here y < x are consecutive occurrences of the color a. By I2A there are exactly two elements u, v ∈ (y, x) with colors adjacent to a. By AC all elements greater than y in I with colors adjacent to a are in (y, x). Hence u and v are the only such elements. This shows both u and v are in Now suppose a ∼ b. Again start with the case P b ∩ I = ∅. Here P b ∩ (I + x) = ∅ as well. We know Note that there is some color c ∼ b such that there are infinitely many elements less than y in P c ∩ F if and only if the same statement is true for F − x. Thus whether or not such a color c exists we have ψ Otherwise for a ∼ b we have the case P b ∩ I ∅. Here P b ∩ (I + x) ∅ as well. Let z ∈ P b ∩ I. Note z < x by AC and that [z, x] and P b ∩ [z, x] are finite. So P b ∩ I has a maximal element y. Note that x satisfies all three criteria to be in Υ b (F − x, I + x). But x Υ b (F, I) since x I. Observe that y is also maximal Note that there is some color c ∼ b such that there are infinitely many elements greater than y in P c ∩ I if and only if the same statement is true for I + x. Thus whether or not such a color c exists we have Finally suppose a b. Again start with the case P b ∩ I = ∅. Here P b ∩ (I + x) = ∅ as well. An element . If no such minimal element exists, then Since a b, there is a color c ∼ b such that there are infinitely many elements less than y in P c ∩ F if and only if the same statement is true for P c ∩ (F − x). Thus whether or not such a color c exists we have Otherwise for a b we have the case So {µ a } a∈Γ is a component weight function. Since I2A implies I3ND, we can apply Corollary 6.6(b) to get the last statement.

Existence and uniqueness for sl 2 weights along color strings
The actions of the {h a } a∈Γ in a minuscule representation of a semisimple Lie algebra have certain values along their "sl 2 strings." To obtain upper P-minuscule representations of b ′ + and P-minuscule representations of g ′ , we need component weight functions that have these values along the "color strings" of F I(P). The next result is the first step toward obtaining these values. This existence result motivates the properties AC and I2A from a Lie representation viewpoint. Proof. We first show (iii) implies (i). Create a component weight function {η a } a∈Γ that satisfies (iii). Let b ∈ Γ and (F, I) ∈ F I(P). Suppose b is the color of a minimal element y of F. Then η b (F − y, I + y) = 1 by (iii) since y is maximal in I + y. Equation (6.2) gives η b (F − y, I + y) − η b (F, I) = 2. Hence we get η b (F, I) = η b (F − y, I + y) − 2 = −1, yielding (i). Dualize to get (i) implies (iii). This also shows that one choice will work for both Parts (i) and (iii).
We next show (iii) implies (ii). Continue to consider the {η a } a∈Γ above. Let x and y be incomparable elements in P. Define a := κ(x) and b := κ(y). Let F be the filter generated by x and y and set I := P − F.
Note that b is the color of a maximal element of both I +y and I +y+ x. Thus by (iii) we have η b (F −y, I +y) = we can apply Equation (6.2) to also obtain η b (F − y − x, I + y + x) − η b (F − y, I + y) = θ ab . Thus θ ab = 0, and so a b. Thus we get both EC and AC. Now let b ∈ Γ and let x < y be consecutive occurrences of the color b. Define I ′ to be the principal ideal generated by y. Define an ideal I to be I ′ − (x, y], where (x, y] := {z ∈ P | x < z ≤ y}. Also note that x is maximal in I. Define F ′ := P − I ′ and F := P − I. Since (x, y] is finite, the splits (F ′ , I ′ ) and (F, I) are in the same component of F I(P). Since y is maximal in I ′ and x is maximal in I, we have η b (F ′ , I ′ ) = 1 = η b (F, I). Also note I ′ − I = (x, y] and I − I ′ = ∅. Thus using Equation (6.1) we get Since κ(y) = b this equation can be rewritten 2 = c∼b |P c ∩ (x, y)|. Thus I2A holds.

Frontier census properties and eigenvalue bounds
Here we introduce our last coloring properties; they limit the eigenvalues of the coroot actions. For each k ≥ 1 we define two frontier census properties: (MxkGA): For every color a ∈ Γ: If x is maximal in P a , then there are at most k elements greater than x that have their colors adjacent to a, (MnkLA): For every color a ∈ Γ: If x is minimal in P a , then there are at most k elements less than x that have their colors adjacent to a.
We also introduce two more general such properties: (MxFGA): For every color a ∈ Γ: If x is maximal in P a , then the number of elements greater than x that have their colors adjacent to a is finite, (MnFLA): For every color a ∈ Γ: If x is minimal in P a , then the number of elements less than x that have their colors adjacent to a is finite.
The properties Mx1GA and Mn1LA are the most important of these properties; in [PrSt] the property Mn2LA will also be used. In Section 13 we indicate how Mx1GA and Mn1LA revamp, generalize, and unify axioms considered by Stembridge and Green. The property Mx1GA was retrospectively found to be implicitly present in Proposition 2.5 of [Ste]. That early statement in [Ste] was formulated in terms of decompositions of Weyl group elements w, before the heap finite colored posets were introduced.
The three facts below summarize the interactions between the frontier census properties and the component weight function {µ a } a∈Γ constructed in Section 7. Define E µ := {µ a (F, I) | a ∈ Γ, (F, I) ∈ F I(P)}. Since these facts will not be cited below, their proofs are relegated to [Str]. We can also say when these bounds for E µ are attained: 10 Upper P -minuscule representations of b ′ + The first main result of this paper gives necessary and sufficient conditions on coloring properties for P so that F I(P) carries an upper P-minuscule representation of b ′ + ; this notion was defined at the end of Section 4.
Theorem 10.1. Let P be a poset whose elements are colored by the nodes of a finite simple graph Γ. Let When either of these conditions is satisfied, the µ-diagonal operators {M a } a∈Γ of Section 7 can be used to give the actions of the {h a } a∈Γ .
It can be confirmed that the poset displayed in Figure 4.1 satisfies the Mx1GA version of Condition (ii); the poset displayed in Figure 2.1 satisfies both versions. Once the classification of posets satisfying this collection of properties has been obtained in [PrSt], for connected P it can be shown that the µ-diagonal operators are the unique operators satisfying Part (i) here (after any trivial components in F I(P) have been discarded).
Proof. Assume that (i) holds for b ′ + . Since this representation is X-square nilpotent, by Theorem 6.8 we know P satisfies EC and NA. Let {H a } a∈Γ be the diagonal operators for this representation with h ′ -weight {η a } a∈Γ and eigenvalue set E η := {η a (F, I) | a ∈ Γ, (F, I) ∈ F I(P)}. By Corollary 6.6(a) we know that Since y is maximal in P b we have |P b ∩ ((y, u] ∪ (y, v])| = 0. Since u and v have colors adjacent to b, we have Since y is maximal in I and κ(y) = b, by the penultimate statement of Proposition 8.1 we get η b (F, I) = 1. Thus the inequality becomes η b (F ′ , I ′ ) ≤ −1. By (10.1) we have η b (F ′ , I ′ ) ≥ −1, and so η b (F ′ , I ′ ) = −1. So by (10.2) we know that F ′ has a minimal element z of color b. Since y ∈ I ′ , by EC we have y < z. But y is maximal in P b , so this is a contradiction. Now assume that (ii) holds. Since P satisfies EC, AC, and I2A, we know by Proposition 7.1 that {µ a } a∈Γ is a component weight function. We use the operators {M a } a∈Γ specified by {µ a } a∈Γ to get the desired actions of the {h a } a∈Γ . Using EC, NA, and I3ND (which is implied by I2A), Corollary 6.6(b) says F I(P) carries an X-square nilpotent representation of b ′ + when using the {M a } a∈Γ . The eigenvalue set E µ of these operators is contained in Z. Let b ∈ Γ and (F, I) ∈ F I(P). For the sake of contradiction, suppose µ b (F, I) < −1. Then necessarily µ b (F, I) = 1 − υ b (F, I) and υ b (F, I) ≥ 3. Thus P b ∩ I must have a maximal element y of color b, and there must be three or more elements greater than y in I with colors adjacent to b. Then by AC and I2A we see P b ∩ F = ∅, so y is maximal in P b . But this would violate Mx1GA. Thus we see that µ b (F, I) ≥ −1, and so E µ ⊆ {−1, 0, 1, 2, . . . }. The last statement of Proposition 8.1 says µ b (F, I) = −1 if b is the color of a minimal element of F.
To finish, we need to confirm the "only if" direction of Part (ii) of the definition of upper P-minuscule.
This shows P b ∩ I has a maximal element y since υ b (F, I) = 1 otherwise. Let c ∈ Γ be such that c ∼ b. If there are infinitely many elements greater than y in P c ∩ I, then P b ∩ F is empty by AC and local finiteness.
But then y would be maximal in P b , which would violate Mx1GA. Thus |Υ b (F, I)| = υ b (F, I) = 2. So there are two distinct elements u and v in I greater than y with colors adjacent to b. By Mx1GA we know that y cannot be maximal in P b . Let z ∈ P b be such that y < z are consecutive occurrences of the color b. Since y is maximal in P b ∩ I, we have z ∈ F. By AC we know u, v ∈ (y, z). Suppose w ∈ P with w → z. Then NA implies that κ(w) ∼ b. Thus by AC we have w ∈ (y, z). By I2A we know that w = u or w = v, so w ∈ I. Thus z is minimal in F and has color b.
Otherwise we have the case shows P b ∩ F has a minimal element y since ψ b (F, I) = 1 otherwise. Suppose w ∈ P with w → y. Again by NA we know that κ(w) ∼ b. If there is some color c ∼ b such that there are infinitely many elements less than y in P c ∩ F, then ψ b (F, I) ≥ 1. Since ψ b (F, I) = 0, there are only finitely many elements less than y in Dualize to obtain the analogous equivalence for b ′ − .
The following fact is used for Corollary 10.3 and Theorem 11.2; it is confirmed in [Str]: We can now obtain the equivalence of three of the conditions in Theorem 11.2 below: The conditions in (iv) completely specify the operators, so they are unique. The µ-diagonal operators satisfy (ii) when these conditions hold, and this proof showed that any diagonal operators satisfying (ii) also satisfy (iv).

P -minuscule representations of g ′
We obtain our foremost main result, Theorem 11.2. It provides necessary as well as sufficient conditions on coloring properties for P needed for F I(P) to carry a P-minuscule representation of g ′ . Both its statement and its proof simultaneously handle finite and infinite posets.
Lemma 11.1. Suppose P satisfies EC and ND. The relation [X b , Y a ] = 0 holds when a, b ∈ Γ are distinct.
Proof. Fix distinct a, b ∈ Γ and let (F, I) ∈ F I(P). Suppose that X b Y a . F, I 0. Then there are distinct elements x, y ∈ P such that y is maximal in I and x is minimal in F + y with κ(y) = a and κ(x) = b such that X b Y a . F, I = (F + y) − x, (I − y) + x . Since x y and x is minimal in F + y, we see x is also minimal in F. Since x and y are minimal elements of the same filter, they are incomparable. Thus we have Y a ]. F, I = 0. Dualize to handle the case Y a X b . F, I 0.
Here we characterize the P-minuscule representations of g ′ in several ways; see Section 3 for the definitions of X-and Y-square nilpotent actions and Section 4 for the definitions of (upper and lower) P-minuscule representations of g ′ (and b ′ + and b ′ − ). In Section 14 we compare the implication (v)⇒(i) to Theorem 4.1.6(i) of [Gr3].
Theorem 11.2. Let P be a poset whose elements are colored by the nodes of a finite simple graph Γ. Let This proof showed that the diagonal operators {H a } a∈Γ satisfying (iii) also satisfy the conditions of (iv).
Hence the diagonal operators {H a } a∈Γ for (i) also satisfy (iv). Corollary 10.3 showed that the µ-diagonal operators are the unique operators satisfying the conditions of (ii) or (iv).
12 Γ-colored d-complete and Γ-colored minuscule posets We say a locally finite poset P (of any cardinality) colored with a finite simple graph Γ is a Γ-colored dcomplete poset if it satisfies EC, NA, AC, I2A, and Mx1GA with respect to Γ.

Identifications and classifications, axiom comparisons
With Proctor in [PrSt] we show that any Γ-colored minuscule (respectively Γ-colored d-complete) poset is the direct sum of connected Γ k -colored minuscule (respectively Γ k -colored d-complete) posets, each of which may be finite or infinite, on the p ≥ 1 connected components Γ 1 , . . . , Γ p of Γ. We then identify three of the four classes of such connected posets in our conceptual 2 × 2 table as previously studied (and classified) classes of posets. This enables us to classify all Γ-colored minuscule and Γ-colored d-complete posets: Theorem 13.1. Classifications for connected finite cases.
(a) The connected finite Γ-colored minuscule posets are exactly the colored irreducible minuscule posets of Theorem 11 of [Pr1] for the simply laced Dynkin diagrams. Hence the connected finite Γ-colored minuscule posets are the simply laced cases in the list of colored minuscule posets in [Pr1].
(b) The connected finite Γ-colored d-complete posets are exactly the connected colored d-complete posets of [Pr3] with top trees denoted by Γ. Hence the connected finite Γ-colored d-complete posets form the list of connected d-complete posets in [Pr2], once those have been uniquely colored as in [Pr3] by the nodes of their top trees.
The poset P displayed in Figure 2.1 is a full heap; this is defined below.
Theorem 13.2. Classifications for connected infinite cases.
(a) The connected infinite Γ-colored minuscule posets are exactly the connected full heaps of [Gr3] for the simply laced Dynkin diagrams. So by Theorem 4.7.1 of [McG] the connected infinite Γ-colored minuscule posets are the simply laced cases in the list of full heaps in Theorem 6.6.2 of [Gr3].
(b) The connected infinite Γ-colored d-complete posets are exactly the filters of the connected full heaps of [Gr3] for the simply laced Dynkin diagrams. Hence the connected infinite Γ-colored d-complete posets are the filters of the simply laced cases in the list of full heaps in Theorem 6.6.2 of [Gr3].
It is not easy to see that a connected infinite Γ-colored minuscule (d-complete) poset is a (filter of) a full heap: In [PrSt] we will first use the weight function {µ a } a∈Γ to show that if Γ is connected and each color set P a in a Γ-colored d-complete poset is unbounded above, then P satisfies Mn2LA. The easy proof that full heaps are Γ-colored minuscule is outlined below.
Having the new (necessary) direction of Theorem 11.2 available will allow us to combine two major results of Green and of McGregor-Dorsey. This produces a classification of the P-minuscule representations of g ′ . We will use Theorem 10.1 to also classify the upper P-minuscule representations of b ′ + : Theorem 13.3. For simply laced graphs Γ and connected posets P we have: To compare axioms, restrict [Ste] and [Gr3] to their simply laced cases. Let NWA indicate that the colors of two neighbors must be equal or adjacent. In the presence of I2A the property NWA becomes NA. We compare our properties to H1-H4 in [Ste], which define "dominant minuscule heap." Note that the collection EC, AC, and NWA is his H1 and our I2A is his H2. Hence our collection EC, NA, AC, and I2A is equivalent to his collection H1 and H2. When P is finite our Mx1GA can be shown to be equivalent to his collection H3 and H4. So our finite Γ-colored d-complete posets constitute the simply laced case of those H1-H4 heaps. Next we compare our properties to H1-H2 and F1-F3 in [Gr3], which define "full heap" for locally finite posets. Our EC and AC is his H1 and our I2A is his F3. By his Lemma 2.1.5(iii) and Exercise 2.1.13(ii), one can see that NWA is his H2. Hence our collection EC, NA, AC, and I2A is equivalent to his collection H1, H2, and F3. His F1 requires that P a Z for all a ∈ Γ. His F2 requires that if a, b ∈ Γ with a ∼ b and x ∈ P a , then x has a neighbor in P b . It could have been seen in [Gr3] that the collection H1, H2, F1, and F3 implies F2; details will be given in [PrSt]. Both Mx1GA and Mn1LA are vacuously satisfied when F1 holds. So each full heap over Γ is an infinite Γ-colored minuscule poset.
The properties Mx1GA and Mn1LA allow us to give uniform definitions of Γ-colored d-complete and Γcolored minuscule posets that unify (what could be referred to as) the finite d-complete posets of [Ste] with the infinite minuscule posets of [Gr3]: Omitting Mn1LA from the Γ-colored minuscule list of properties creates the Γ-colored d-complete list of properties. Our definitions work for both the finite and infinite cases because Mx1GA (and Mn1LA) is (are) vacuously satisfied when the P b are unbounded above (and below) for all b ∈ Γ.
14 Representation remarks, abstract minuscule representations Let g ′ be a derived Kac-Moody algebra with simply laced Dynkin diagram Γ. Let P be a connected full heap over Γ. Let ϕ 1 be the representation of g ′ constructed in Theorem 4.1.6(i) of [Gr3]. By Theorem 13.2 we know that P arises as a Γ-colored minuscule poset. The implication (v)⇒(i) of Theorem 11.2 shows P can be used to build a P-minuscule representation ϕ 2 of g ′ . Here Green's representation ϕ 1 of g ′ can be obtained from our P-minuscule representation ϕ 2 of g ′ by removing the splits (P, ∅) and (∅, P) from our F I(P).
Can these full heap representations be extended from g ′ to the full Kac-Moody algebra g for Γ? For full heaps of untwisted affine type, Theorem 7.10 of [Gr1] did this. By Theorem 4.7.1 of [McG], every simply laced full heap is of this type. Here we indicate how to extend Theorem 11.2 from g ′ to g. We say F I(P) carries a representation of g if a representation of g ′ carried by F I(P) can be extended to a representation of g. Such a representation is P-minuscule if the restriction to g ′ is P-minuscule.
Theorem 14.1. Let P be a poset whose elements are colored by the nodes of a finite simple graph Γ. Let F I(P) be the lattice of filter-ideal splits of P. The following are equivalent: (i) The lattice F I(P) carries a P-minuscule representation of g.
(ii) The poset P is a Γ-colored minuscule poset.
We outline the proof: Restrict from g to g ′ and apply Theorem 11.2 to see that (i) implies (ii). Now let P be Γ-colored minuscule; for this outline assume P is connected. Start by applying Theorem 11.2 to obtain a P-minuscule representation of g ′ . Then use Theorem 13.3: If P is finite, then it must be one of the minuscule posets of [Pr1] and so g = g ′ . If P is infinite, then it is a full heap. Extend from g ′ to g using Theorem 7.10 of [Gr1].
For combinatorial simplicity, here we have not considered representing the derived Kac-Moody algebras g ′ arising from non-simply laced Dynkin diagrams Γ. For this case, most of our combinatorial definitions of Section 2 need to be extended. After enlarging the sets of defining relations for n ± , b ′ ± , and g ′ , none of the algebraic and combined definitions of Sections 3 and 4 need to be changed. In particular, the definitions of upper P-minuscule representations of b ′ + and P-minuscule representations of g ′ do not need to be changed. The coloring properties I3ND and I2A and the frontier census properties MxkGA and MnkLA must also be extended; for example, the property I2A needs to be extended to H2 of [Ste]. Once these adjustments are made, we can extend [Str] our results.
The upper P-minuscule representations of b ′ + in Theorem 10.1 for finite Γ-colored d-complete posets P can be shown to be exactly the restrictions to b ′ + of the Demazure b + -modules for the dominant minuscule Weyl group elements. To outline one direction, let P be such a poset. By the equivalence between finite Γ-colored d-complete posets and Stembridge's dominant minuscule heaps [Ste], we know P is the heap of some dominant minuscule element w of the Weyl group of Γ. Thus there is a dominant integral weight λ of g such that w is λ-minuscule. The order dual of the Bruhat interval [e, w] is isomorphic to F I(P).
(One could instead use Theorem B of [Pr3] here.). Let V(λ) be an integrable highest weight g-module with highest weight λ, and let V w be the Demazure module [Kum,Def. 8.1.22] associated to w. Since w is λminuscule, all of the "reflection multiplicities" are unity when the elements in (the dual of) the interval [e, w] are generated upwardly from w. So it can be seen that the weights for V w are of the form v.λ for v ∈ [e, w] and hence have multiplicity 1. Then the upper P-minuscule representation of b ′ + from Theorem 10.1 is isomorphic to the restriction of V w from b + to b ′ + . Conversely, given such a Demazure b + -module V w for a dominant minuscule w, it can be shown that the heap P of w (which is a finite Γ-colored d-complete poset) produces an upper P-minuscule representation of b ′ + to which the restriction of V w to b ′ + is isomorphic. A weight representation of a Kac-Moody algebra g is one for which the actions of the elements of its Cartan subalgebra h can be simultaneously diagonalized. It is multiplicity free if each of the resulting h-weight spaces is 1-dimensional. The weights inherit the usual simple root partial order from h * . An irreducible highest weight representation of a semisimple Lie algebra g is minuscule if each of its weights is in the Weyl group orbit of the highest weight. No such highest weight representations exist when g is infinite dimensional. Is it nonetheless possible to develop a meaningful notion of "minuscule" representation for the infinite dimensional Kac-Moody algebras? This paper is the first in a series of three papers in which we endeavor to develop such a notion and to classify all such representations. Green's doubly infinite representations of certain affine Lie algebras built from full heaps are the prime candidates to be regarded as the minuscule representations of the infinite-dimensional Kac-Moody algebras. These representations and the minuscule representations of the semisimple Lie algebras are multiplicity free weight representations, their h-weights form a connected poset under the order on h * , and their h ′ -weights have values only in the set {−1, 0, 1} for the simple coroot actions. We propose that these three axioms constitute the definition of a "minuscule" representation of any Kac-Moody algebra. We hope to show that when such a representation is restricted to g ′ it is essentially a P-minuscule representation carried by the lattice F I(P) of splits for a Γ-colored poset P. We could then apply the new "necessary" direction of the foremost main result of this paper, Theorem 11.2, to see that P is a Γ-colored minuscule poset. Then Theorem 13.3(a) would be used to list all such representations.