Séminaire Lotharingien de Combinatoire 75 (2016), Article B75b

HOMOGENEOUS REPRESENTATIONS OF TYPE A KLR-ALGEBRAS AND DYCK PATHS

GABRIEL FEINBERG AND KYU-HWAN LEE^{⋄}

Abstract. The Khovanov–Lauda–Rouquier (KLR) algebra arose out of attempts to categorify quantum groups. Kleshchev and Ram proved a result reducing the representation theory of these algebras of finite type to the study of irreducible cuspidal representations. In type A, these cuspidal representations are included in the class of homogeneous representations, which are related to fully commutative elements of the corresponding Coxeter groups. In this paper, we study fully commutative elements using combinatorics of Dyck paths. Thereby we classify and enumerate the homogeneous representations for KLR algebras of types A and obtain a dimension formula for some of these representations from combinatorics of Dyck paths.

Introduction

Introduced by Khovanov and Lauda [11] and independently by Rouquier [16], the Khovanov–Lauda–Rouquier (KLR) algebras (also known as quiver Hecke algebras) have been the focus of many recent studies. In particular, these algebras categorify the lower (or upper) half of a quantum group. More precisely, the Cartan datum associated with a Kac–Moody algebra 𝔤 gives rise to a KLR algebra R. The category of finitely generated projective graded modules of this algebra can be given a bialgebra structure by taking the Grothendieck group, and taking the induction and restriction functors as multiplication and co-multiplication. To say that the KLR algebra R categorifies the negative part U_{q}^{−}(𝔤) of the quantum group, is to say that this bialgebra is isomorphic to Lusztig’s integral form of U_{q}^{−}(𝔤).

In the paper [13], Kleshchev and Ram significantly reduce the problem of describing the irreducible representations of the finite type KLR algebras. They defined a class of cuspidal representations for these algebras, and showed that every irreducible representation appears as the head of some induction of these cuspidals, and constructed almost all cuspidal representations. Hill, Melvin, and Mondragon in [8] completed the construction of cuspidals in all finite types, and re-frame them in a more unified manner.

2010 Mathematics Subject Classification. Primary 16G99; Secondary 05E10.

⋄This work was partially supported by a grant from the Simons Foundation (#318706).

Furthermore, Lauda and Vazirani imposed a crystal structure on the isomorphism classes of irreducible representations of a KLR algebra. They showed in [14] that this crystal is isomorphic to the crystal B(∞) of the quantum group U_{q}(𝔤). Crystals are also used by Benkart, Kang, Oh, and Park in [1] to give a new approach towards the construction of irreducible representations. For more background and other developments, see [4] and [10].

In the process of constructing the cuspidal representations, Kleshchev and Ram defined a class of representations known as homogeneous representations [12], those that are concentrated in a single degree. Homogeneous representations include most of the cuspidal representations for finite types with a suitable choice of ordering on words. Therefore it is important to completely understand these representations. As shown in [12], homogeneous representations can be constructed from the sets of reduced words of fully commutative elements in the corresponding Coxeter group. These elements were studied by Fan [7] and Stembridge [17, 18], and are closely related to Temperley–Lieb algebras [9].

Motivated by this connection to the homogeneous representations of KLR algebras, we study, in this paper, fully commutative elements of the Coxeter groups of type A_{n}. We decompose the set of fully commutative elements into natural subsets according to the lengths of fully commutative elements, and study combinatorial properties of these subsets. Our main result (Theorem 2.1) shows that the fully commutative elements of a given length k can be parameterized by the Dyck paths of semi-length n with the property that (sum of peak heights) − (number of peaks) = k. The main idea of the proof is to investigate a canonical form of reduced words for fully commutative elements.^{1}

After the parameterization is obtained, we classify and enumerate the homogeneous representations of KLR algebras of type A according to the decomposition of the set of fully commutative elements (Corollary 2.2). In their paper [12], Kleshchev and Ram gave a parameterization of homogeneous representations using skew shapes. Our result uses different combinatorial objects, i.e., Dyck paths, and gives a refinement of the classification. Furthermore, we obtain a dimension formula for some homogeneous representations using combinatorics of Dyck paths (Proposition 3.2), which is a reformulation of the Peterson– Proctor formula. The precise relationship between skew shapes and Dyck paths is not clear at the present; our Dyck path realization is more directly related to a canonical form of fully commutative elements.

1After this paper was accepted, the authors were informed by C. Krattenthaler that the problem of enumerating fully commutative elements by length was also studied by Biagioli, Jouhet and Nadeau in [2]. Interestingly, they give a bijection to Motzkin paths with two types of horizontal steps. Since there is a well-known bijection of these Motzkin paths to Dyck paths, one obtains a different bijection from fully commutative elements to Dyck paths, where the length of a fully commutative element is given by half of the sum of the heights of points of the corresponding Dyck path at even positions. Then, by combining our result with their result and by going through the fully commutative elements, we obtain a bijection from Dyck paths to Dyck paths, which is compatible with the two different ways to parameterize fully commutative elements with respect to length.

The outline of this paper is as follows. In Section 1, we fix notations, briefly review the representations of KLR algebras, and explain the relationship between homogeneous representations and fully commutative elements of a Coxeter group. In Section 2, we introduce Dyck paths and study a canonical form of reduced words of fully commutative elements and obtain the main results of this paper. In Section 3, we prove a dimension formula for homogeneous representations when the corresponding Dyck paths satisfy a certain condition.

Acknowledgments. Part of this research was performed while both authors were visiting the Institute for Computational and Experimental Research in Mathematics (ICERM) during the spring of 2013 for the special program “Automorphic Forms, Combinatorial Representation Theory and Multiple Dirichlet Series”. They wish to thank the organizers and staffs.

1. KLR Algebras and Homogeneous Representations

1.1. Definitions. To define a KLR algebra, we begin with a quiver Γ. In this paper, we will focus mainly on quivers of Dynkin types A_{n}, but for the definition, any finite quiver with no double bonds will suffice. Let I be the set indexing the vertices of Γ, and for indices i ≠ j, we will say that i and j are neighbors if i → j or i ← j. Define Q_{+} = ⨁_{i∈I} ℤ_{≥0} α_{i} as the non-negative lattice with basis {α_{i} : i ∈ I}. The set of all words in the alphabet I is denoted by ⟨I⟩, and for a fixed α = ∑_{i∈I} c_{i}α_{i} ∈ Q_{+}, let ⟨I⟩_{α} be the set of words w on the alphabet I such that each i ∈ I occurs exactly c_{i} times in w. We define the height of α to be ∑_{i∈I} c_{i}. We will write w = [w_{1}, w_{2}, …, w_{d}], w_{j} ∈ I.

Now, fix an arbitrary ground field 𝔽 and choose an element α ∈ Q_{+}. Then the Khovanov– Lauda–Rouquier algebra R_{α} is the associative 𝔽-algebra generated by:

• idempotents {e(w) | w ∈ ⟨I⟩_{α}},

• symmetric generators {ψ_{1}, …, ψ_{d−1}} where d is the height of α,
• polynomial generators {y_{1}, …, y_{d}},

subject to the relations

e(w)e(v) = δ_{wv}e(w),  ∑_{w∈⟨I⟩_{α}} e(w) = 1; (1.1)

y_{k}e(w) = e(w)y_{k}; (1.2) ψ_{k}e(w) = e(s_{k}w)ψ_{k}; (1.3) y_{k}y_{ℓ} = y_{ℓ}y_{k}; (1.4) y_{k}ψ_{ℓ} = ψ_{ℓ}y_{k} for k ≠ ℓ, ℓ + 1; (1.5)

(1.6) (y_{k+1}ψ_{k} − ψ_{k}y_{k})e(w) = \begin{cases}e(w), & if w_{k} = w_{k+1}, \\ 0, & otherwise; \end{cases}

(1.7) (ψ_{k}y_{k+1} − y_{k}ψ_{k})e(w) = \begin{cases}e(w), & if w_{k} = w_{k+1}, \\ 0, & otherwise; \end{cases}

(1.8) ψ_{k}^{2}e(w) = \begin{cases}0, & if w_{k} = w_{k+1}, \\ (y_{k} − y_{k+1})e(w), & if w_{k} → w_{k+1}, \\ (y_{k+1} − y_{k})e(w), & if w_{k} ← w_{k+1}, \\ e(w), & otherwise; \end{cases}

ψ_{k}ψ_{ℓ} = ψ_{ℓ}ψ_{k} for |k − ℓ| > 1; (1.9)

(1.10) (ψ_{k+1}ψ_{k}ψ_{k+1} − ψ_{k}ψ_{k+1}ψ_{k})e(w) = \begin{cases}e(w), & if w_{k+2} = w_{k} → w_{k+1}, \\ −e(w), & if w_{k+2} = w_{k} ← w_{k+1}, \\ 0, & otherwise. \end{cases}

Here δ_{wv} in (1.1) is the Kronecker delta and, in (1.3), s_{k} is the k^{th} simple transposition in the symmetric group S_{d}, acting on the word w by swapping the letters in the k^{th} and (k + 1)^{st} positions. If Γ is a Dynkin-type quiver, we will say that R_{α} is a KLR algebra of that type.

We impose a ℤ-grading on R_{α} by

deg(e(w)) = 0, deg(y_{i}) = 2, (1.11)

(1.12) deg(ψ_{i}e(w)) = \begin{cases}−2, & if w_{i} = w_{i+1}, \\ 1, & if w_{i}, w_{i+1} are neighbors in Γ, \\ 0, & if w_{i}, w_{i+1} are not neighbors in Γ. \end{cases}

Set R = ⨁_{α∈Q_{+}} R_{α}, and let Rep(R) be the category of finite dimensional graded Rmodules, and denote its Grothendieck group by [Rep(R)]. Then Rep(R) categorifies one half of the quantum group. More precisely, let f and ′f be Lusztig’s algebras defined in [15, Section 1.2] attached to the Cartan datum encoded in the quiver Γ over the field ℚ(v). We put q = v^{−1} and A = ℤ[q, q^{−1}], and let ′f_{A} and f_{A} be the A-forms of ′f and f, respectively. Consider the graded duals ′f^{∗} and f^{∗}, and their A-forms

′f_{A}^{∗} := {x ∈ ′f^{∗} : x(′f_{A}) ⊂ A} and f_{A}^{∗} := {x ∈ f^{∗} : x(f_{A}) ⊂ A}.

Then Khovanov and Lauda [11] prove that there is an A-linear (bialgebra) isomorphism [Rep(R)] ⟶^{∼} f_{A}^{∗}. More details can be found in [11, 13].

A word w ∈ ⟨I⟩_{α} is naturally considered as an element of ′f_{A}^{∗} which is dual to the corresponding monomial in ′f_{A}. Let M be a finite dimensional graded R_{α}-module. Define the q-character of M by

ch_{q} M := ∑_{w∈⟨I⟩_{α}}(dim_{q} M_{w}) w ∈ ′f_{A}^{∗},

where M_{w} = e(w)M and dim_{q} V := ∑_{n∈ℤ}(dim V_{n}) q^{n} ∈ A for V = ⨁_{n∈ℤ} V_{n}. A nonempty word w is called Lyndon if it is lexicographically smaller than all its proper right factors with respect to a fixed total ordering on I. For x ∈ ′f^{∗} we denote by max(x) the largest word appearing in x. A word w ∈ ⟨I⟩ is called good if there is x ∈ f^{∗} such that w = max(x). Given a module L ∈ Rep(R_{α}), we say that w ∈ ⟨I⟩ is the highest weight of L if w = max(ch_{q} L). An irreducible module L ∈ Rep(R_{α}) is called cuspidal if its highest weight is a good Lyndon word.

The following theorem explains the importance of cuspidal representations as building blocks for all irreducible representations of R_{α}.

Theorem 1.1 ([13]; [8], 4.1.1). Assume that Γ is of finite Dynkin type. Then any irreducible graded R_{α}-module for α ∈ Q_{+} is given by an irreducible head of a standard representation induced from cuspidal representations up to isomorphism and degree shift.

1.2. Homogeneous representations. We define a homogeneous representation of a KLR algebra to be an irreducible, graded representation fixed in a single degree (with respect to the ℤ-grading described in (1.11) and (1.12)). Homogeneous representations form an important class of irreducible modules since most of the cuspidal representations are homogeneous with a suitable choice of ordering on ⟨I⟩ ([13, 8]). After introducing some terminology, we will describe these representations in a combinatorial way. We continue to assume that Γ is a simply-laced quiver.

Fix an α ∈ Q_{+} and let d be the height of α. For any word w ∈ ⟨I⟩_{α}, we say that the simple transposition s_{r} ∈ S_{d} is an admissible transposition for w if the letters w_{r} and w_{r+1} are neither equal nor neighbors in the quiver Γ. Following Kleshchev and Ram [12], we define the weight graph G_{α} with vertices given by ⟨I⟩_{α}. Two words w, v ∈ ⟨I⟩_{α} are connected by an edge if there is an admissible transposition s_{r} such that s_{r}w = v.

We say that a connected component C of the weight graph G_{α} is homogeneous if the following property holds for every w ∈ C:

(1.13) If w_{r} = w_{s} for some 1 ≤ r < s ≤ d, then there exist t, u with r < t < u < s such that w_{r} is a neighbor of both w_{t} and w_{u}.

A word satisfying condition (1.13) will be called a homogeneous word.

A main theorem of [12] shows that the homogeneous components of G_{α} exactly parameterize the homogeneous representations of the KLR algebra R_{α}.

Theorem 1.2 ([12], Theorem 3.4). Let C be a homogeneous component of the weight graph G_{α}. Define an 𝔽-vector space S(C) with basis {v_{w} : w ∈ C} labeled by the vertices in C. Then we have an R_{α}-action on S(C) given by

e(w′)v_{w} = δ_{w,w′}v_{w},  w′ ∈ ⟨I⟩_{α}, w ∈ C,
y_{r}v_{w} = 0,  1 ≤ r ≤ d, w ∈ C,
ψ_{r}v_{w} = \begin{cases}v_{s_{r}w}, & if s_{r}w ∈ C, \\ 0, & otherwise, \end{cases} 1 ≤ r ≤ d − 1, w ∈ C,

which gives S(C) the structure of a homogeneous, irreducible R_{α}-module. Further S(C) ≇ S(C′) if C ≠ C′, and this construction gives all of the irreducible homogeneous modules, up to isomorphism.

As a result, the task of identifying homogeneous representations of a KLR algebra is reduced to identifying homogeneous components in a weight graph. This is simplified further by the following lemma.

Lemma 1.3 ([12], Lemma 3.3). A connected component C of the weight graph G_{α} is homogeneous if and only if an element w ∈ C satisfies the condition (1.13).

Recall that we call a word satisfying condition (1.13) a homogeneous word. The homogeneous words have other combinatorial characterizations, which we explore in the next subsection.

1.3. Fully commutative elements of Coxeter groups. Since the homogeneity of w ∈ ⟨I⟩ does not depend on the orientation of a quiver, it is enough to consider Dynkin diagrams and the corresponding Coxeter groups. Given a simply laced Dynkin diagram, the corresponding Coxeter group will be denoted by W and the generators by s_{i}, i ∈ I. A reduced expression s_{i_{1}} ⋯ s_{i_{r}} will be identified with the word [i_{1}, …, i_{r}] in ⟨I⟩. The identity element will be identified with the empty word [ ].

An element w ∈ W is said to be fully commutative if any reduced word for w can be obtained from any other by interchanges of adjacent commuting generators, or equivalently if no reduced word for w has [i, i′, i] as a subword where i and i′ are neighbors in the Dynkin diagram. Now we have the following lemma, which was first observed by Kleshchev and Ram.

Lemma 1.4 ([12]).

(1) A homogeneous component of the weight graph G_{α} contains as its vertices exactly the set of reduced expressions for a fully commutative element in W.

(2) The set of homogeneous components is in bijection with the set of fully commutative elements in W.

Stembridge [17] classified all of the Coxeter groups that have finitely many fully commutative elements, completing the work of Fan [7], who had done this for the simply-laced types. Fan and Stembridge also enumerated the set of fully commutative elements. In particular, they showed that the number of fully commutative elements in the Coxeter group of type A_{n} is C_{n+1}, where C_{n} is the n^{th} Catalan number, i.e., C_{n} = \frac{1}{n+1} \binom{2n}{n}. This fact has an immediate implication on homogeneous representations by Lemma 1.4.

Corollary 1.5. A KLR algebra R = ⨁_{α∈Q_{+}} R_{α} of type A_{n} has C_{n+1} irreducible homogeneous representations.

In [12], Kleshchev and Ram parameterized homogeneous representations using skew shapes. In this paper, we will decompose the set of fully commutative elements to give a finer enumeration of homogeneous representations in type A_{n}. More precisely, in the next section, our main result is a fine bijection between the family of irreducible homogeneous representations and the set of Dyck paths in accordance with the decomposition of the set of fully commutative elements. This bijection can be used to quickly enumerate the fully commutative elements of a given length and the attached homogeneous representations.

2. Homogeneous Representations of Type A_{n} KLR Algebras

In this section, we describe all of the homogeneous representations of a KLR algebra of type A_{n}, associated with a quiver whose underlying graph is

\begin{matrix}1 \\ 2 \\ n \end{matrix}

We begin by introducing the main combinatorial tool for our study.

2.1. Dyck paths. As in [5], we define a Dyck path as a lattice path in the first quadrant consisting of steps ⟨1, 1⟩ (north-east) and ⟨1, −1⟩ (south-east), beginning at the origin and ending at the point (2n, 0). We refer to n as the semi-length of the path. By a peak we shall mean a rise ⟨1, 1⟩ followed by a fall ⟨1, −1⟩, while a valley is a fall, followed by a rise.

\begin{matrix}(0, 0) \\ (2, 0) \\ (4, 0) \\ (6, 0) \\ (8, 0) \\ (10, 0) \end{matrix}

Figure 2.1. An example of a Dyck path of semilength 5.

We will turn our focus to a particular statistic, k, on these Dyck paths. Denote by D_{n,k} the set of all Dyck paths of semi-length n with the property that

(sum of peak heights) − (number of peaks) = k.

For the example path shown in figure 2.1 we have k = (2 + 3 + 1) − 3 = 3.

Let T (n, k) be the cardinality of the set D_{n,k}, as defined in [6]. It is known that T (n, k) = 0 when k > 1 + ⌊\frac{n^{2}}{4} ⌋. It is convenient to display the sequence of non-zero values as an array with the entry T (n, k) in the n^{th} row from the top (starting with n = 0) and the k^{th} column (beginning with k = 0). The top of the array is shown below.

(2.1)  1 1 1 1 1 2 2 1 3 5 4 1 1 4 9 12 10 4 2 1 5 14 25 31 26 16 9 4 1

It is well known that the number of Dyck paths of semi-length n is equal to the n^{th} Catalan number, C_{n} = \frac{1}{n+1} \binom{2n}{n}, so we have

(2.2)  C_{n} = ∑_{k} T (n, k),

i.e., the sum of entries on the n^{th} row is equal to C_{n}.

Now we consider fully commutative elements in the Coxeter group S_{n+1} and state our main result. Recall that the length of an element of S_{n+1} is defined with respect to the generators (simple transpositions s_{1}, …, s_{n}) of S_{n+1}.

Theorem 2.1. Let C_{n,k} be the set of fully commutative elements of length k in S_{n+1} for k ≥ 0. Then there is a natural bijection Φ : C_{n,k} → D_{n+1,k}. In particular, we have, for k ≥ 0,

|C_{n,k}| = T (n + 1, k).

By Lemma 1.4, we will identify C_{n,k} with the set of homogeneous components of weight graphs G_{α} with α having height k. It follows from Theorem 1.2 that a homogeneous representation is completely determined by a homogeneous component of a weight graph. Thus Theorem 2.1 implies the following results regarding the homogeneous representations.

Corollary 2.2.

(1) There exists a bijection between the irreducible homogeneous representations of the KLR algebra R of type A_{n} and the Dyck paths of semi-length n + 1. Under this bijection, a representation given by a homogeneous component of words with length k corresponds to a Dyck path with

(sum of peak heights) − (number of peaks) = k.

(2) The total number of homogeneous representations of the KLR algebra R of type A_{n}, given by homogeneous words of length k is T (n + 1, k).

We will prove Theorem 2.1 in Section 2.3, after we construct canonical words for fully commutative elements in the next subsection.

2.2. Canonical reduced words. We define the decreasing segments

T_{i}^{j} = \{ \begin{matrix}[j, j − 1, …, i + 1, i], & for i ≤ j, \\ [ ], & for i > j. \end{matrix}

The word T_{i}^{j} will also be considered as the element s_{j}s_{j−1} ⋯ s_{i+1}s_{i} ∈ S_{n+1}. In particular, the product T_{i}^{j}T_{i′}^{j′} given by concatenation is well defined.

These segments will be fundamental, so we record some facts here that we will use freely.

Lemma 2.3. Let T_{i}^{j} be a segment, as defined above. Then we have, for i, i′, j, j′ ∈ I: (1) T_{i}^{j} is a homogeneous word;

(2) if i − 1 = j′ ≥ i′ then T_{i}^{j}T_{i′}^{j′} = T_{i′}^{j};

(3) if j′ < i − 1 then T_{i}^{j}T_{i′}^{j′} = T_{i′}^{j′}T_{i}^{j}.

Proof. These statements follow directly from the definitions. □

We can use these segments to obtain a canonical form for the elements in the Coxeter group S_{n+1}:

Lemma 2.4. Every element in S_{n+1} can be written in the form

(2.3)  T_{i_{1}}^{1}T_{i_{2}}^{2} ⋯ T_{i_{n}}^{n},

where 1 ≤ i_{j} ≤ j + 1 for all 1 ≤ j ≤ n.

Remark 2.5. As a check, notice that there are (n + 1)! choices for the i_{j}’s in this form, and hence (n + 1)! elements in S_{n+1}. The above lemma is standard. One can find a proof in Lemma 3.2 of [3], which uses a Gröbner–Shirshov basis.

Using the canonical form (2.3), we can describe canonical representatives of homogeneous components or fully commutative elements of S_{n+1} in a coherent way.

Proposition 2.6. Every homogeneous component of a weight graph contains a unique word of the form

(2.4)  T_{i_{1}}^{m_{1}}T_{i_{2}}^{m_{2}} ⋯ T_{i_{ℓ}}^{m_{ℓ}}

where i_{j} ≤ m_{j} for each j, 1 ≤ i_{1} < i_{2} < ⋯ < i_{ℓ} ≤ n and m_{1} < m_{2} < ⋯ < m_{ℓ}. Equivalently, a fully commutative element of S_{n+1} can be uniquely written in the form (2.4).

Proof. Clearly, every homogeneous component has a unique word of the form (2.3). After omitting, if any, segments of the form T_{j+1}^{j}, we obtain w = T_{i_{1}}^{m_{1}}T_{i_{2}}^{m_{2}} ⋯ T_{i_{ℓ}}^{m_{ℓ}} with i_{j} ≤ m_{j} for each j and m_{1} < m_{2} < ⋯ < m_{ℓ}. We only need to prove 1 ≤ i_{1} < i_{2} < ⋯ < i_{ℓ} ≤ n. For the sake of contradiction, assume that i_{r} ≥ i_{s} for some r < s. Without loss of generality, suppose that i_{1} ≥ i_{2}. Then w has as a subword [m_{1}, …, i_{1}, m_{2}, …, i_{1}, …, i_{2}]. But this subword has two occurrences of the letter i_{1} separated by only one neighbor i_{1} +1, therefore violating the homogeneity assumption. The equivalence of the second assertion follows from Lemma 1.4. □

2.3. A bijection—proof of Theorem 2.1. Recall that C_{n,k} is the set of fully commutative elements of length k in S_{n+1} for k ≥ 0. By Lemma 1.4, we will also consider C_{n,k} as the set of homogeneous components from all weight graphs G_{α} with α having height k. We need to establish a bijection Φ: C_{n,k} → D_{n+1,k} to prove Theorem 2.1. We first construct a lattice as shown in Figure 2.2, ranging (horizontally) from (0, 0) to (2n + 2, 0). Notice that each square block corresponds to T_{i}^{j} = [j, j − 1, …, i + 1, i] for some i ≤ j, and a Dyck path can have peaks at squares T_{i}^{j} or at bottom triangles. Now suppose that we have a homogeneous component C ∈ C_{n,k}. By Proposition 2.6, we can choose a canonical representative w_{C} = T_{i_{1}}^{m_{1}}T_{i_{2}}^{m_{2}} ⋯ T_{i_{ℓ}}^{m_{ℓ}} with i_{j} ≤ m_{j} for each j, where i_{1} < i_{2} < ⋯ < i_{ℓ} and m_{1} < m_{2} < ⋯ < m_{ℓ}. We write w = w_{C} if there is no peril of confusion.

Definition 2.7. Suppose that C ∈ C_{n,k} and w = T_{i_{1}}^{m_{1}}T_{i_{2}}^{m_{2}} ⋯ T_{i_{ℓ}}^{m_{ℓ}} are as above. Then the Dyck path Φ(C) is defined to be the path with peaks only at the square blocks (see Figure 2.2) containing T_{i_{j}}^{m_{j}} (j = 1, 2, …, ℓ) and possibly, at bottom triangles.

Before we check that the map Φ is well-defined, i.e., Φ(C) ∈ D_{n+1,k}, we consider an example to see how the definition works.

Example 2.8. Suppose the quiver Γ is of type A_{4}, and the homogeneous component C is

(0, 0) (2, 0) (4, 0) (6, 0) (8, 0) (10, 0) (2n, 0) (2n + 2, 0) 1 2 3 4 ⋯ n 21 32 43 · · · · · · 321 432 · · · · · · 4321 · · · · · · · · · · · · n ⋯ 1

Figure 2.2. The triangular lattice for tracing Dyck paths

32143 32413 34213 34231 32431

Then the canonical representative of this component is w = [3, 2, 1, 4, 3] = T_{1}^{3}T_{3}^{4}, and the Dyck path Φ(C) is given by:

(0, 0) (2, 0) (4, 0) (6, 0) (8, 0) (10, 0) 123421 32 43 321 432 4321

Here the sum of peak heights of the Dyck path is 4 + 3 = 7, while the number of peaks is 2. Then we see that k = 7 − 2 = 5 is equal to the length of the corresponding word w = [3, 2, 1, 4, 3].

Lemma 2.9. The map Φ is well-defined.

Proof. Let C ∈ C_{n,k} be a homogeneous component with canonical representative

w = T_{i_{1}}^{m_{1}}T_{i_{2}}^{m_{2}} ⋯ T_{i_{ℓ}}^{m_{ℓ}}.

Since i_{1} < i_{2} < ⋯ < i_{ℓ} and m_{1} < m_{2} < ⋯ < m_{ℓ}, there is no redundancy among the peaks and the corresponding Dyck path D is uniquely determined. Note that each segment T_{i_{j}}^{m_{j}} contains m_{j} − i_{j} + 1 letters. Since w has k letters by assumption, we have

k = ∑_{j=1}^{ℓ}[(m_{j} − i_{j}) + 1] = ℓ + ∑_{j=1}^{ℓ}(m_{j} − i_{j}).

On the other hand, in the path Φ(C), each of the ℓ segments T_{i_{j}}^{m_{j}} corresponds to a peak with height (m_{j} − i_{j}) + 2. We then have

(sum of peak heights) − (# of peaks) = ∑_{j=1}^{ℓ}[(m_{j} − i_{j}) + 2] − ℓ = ℓ + ∑_{j=1}^{ℓ}(m_{j} − i_{j}) = k.

Thus Φ(C) ∈ D_{n+1,k} as desired. □

Definition 2.10. To define the inverse, Ψ : D_{n+1,k} → C_{n,k}, we simply read the words contained in the square blocks of the peaks of the Dyck path D from left to right, ignoring peaks at bottom triangles. Then we obtain w = T_{i_{1}}^{m_{1}}T_{i_{2}}^{m_{2}} ⋯ T_{i_{ℓ}}^{m_{ℓ}}, and w determines the corresponding homogeneous component Ψ(D).

A similar argument as in Lemma 2.9 shows that the map Ψ is well-defined.

Example 2.11. Suppose that we have the Dyck path D:

(0, 0) (2, 0) (4, 0) (6, 0) (8, 0) (10, 0) 123421 32 43 321 432 4321 (0, 0) (2, 0) (4, 0) (6, 0) (8, 0) (10, 0)

Reading, from left to right, the segments contained in the peaks, we see that the component Ψ(D) is represented by the word [2, 4, 3] = T_{2}^{2}T_{3}^{4}. This is the homogeneous component

{243 \atop 423}

Note that the sum of peak heights is 1 + 2 + 3 = 6 and the number of peaks is 3. The value of k = 6 − 3 = 3 equals the length of w = [2, 4, 3].

Now we complete the proof that the map Φ is a bijection with inverse Ψ. It is clear from the construction that the blocks in the lattice that form the peaks of Φ(C) contain the words, T_{i_{1}}^{m_{1}}, T_{i_{2}}^{m_{2}}, …, T_{i_{j}}^{m_{ℓ}}, respectively. Thus, we see that Ψ(Φ(C)) = C. Conversely, suppose that D ∈ D is a Dyck path that has been superimposed on the triangular lattice, and assume that D has ℓ peaks (ℓ > 0) that correspond to square blocks. Reading the words occurring at each of these peaks, we obtain T_{i_{1}}^{m_{1}}, T_{i_{2}}^{m_{2}}, …, T_{i_{ℓ}}^{m_{ℓ}}. We notice that, by construction, i_{1} < i_{2} < ⋯ < i_{ℓ} and m_{1} < m_{2} < ⋯ < m_{ℓ}. By Proposition 2.6, the word T_{i_{1}}^{m_{1}}T_{i_{2}}^{m_{2}} ⋯ T_{i_{ℓ}}^{m_{ℓ}} represents a homogeneous component. We also see that Φ(Ψ(D)) = D, and so Ψ is a two-sided inverse of Φ, proving the bijection. This completes the proof of Theorem 2.1.

3. Dimensions of Homogeneous Representations

In [12], Kleshchev and Ram explain how each fully commutative element w of the Weyl group A_{n} can be associated with an abacus diagram, which gives rise to a skew tableau λ. Further, if w is a dominant minuscule element, the Peterson–Proctor hook formula applied to this tableau will count the number of reduced expressions for the fully commutative element, and thus count the dimension of the corresponding R_{α}-module.

In this section, we will adopt our parameterization of the homogeneous modules and obtain a dimension formula only using combinatorics of Dyck paths. We begin by extending the ascents on the Dyck path to connect peaks of the Dyck path with the corresponding points on the x-axis, and highlighting any square block that appears directly under one of these extended ascents. For example, we have

(0, 0) ; (2, 0) ; (4, 0) ; (6, 0) ; (8, 0) ; (10, 0) ; (12, 0) ; 1 3 5 21 43 321 ; 2 32 4

Figure 3.1. A Dyck path with extended ascents

Now, for any block T_{i}^{j} that appears on an extended ascent, we draw a subpath P_{D}(i, j) according to the following instructions:

(1) Draw a path from the x-axis past blocks T_{i}^{i}, T_{i}^{i+1}, … up to the peak of block T_{i}^{j}.

(2) From there, the path descends until it hits another extended ascent. If the path does not hit any extended ascent, it returns to the x-axis.

(3) If the path hits an extended ascent (before it returns to the x-axis), take one step up, and then go back to step (2).

(4) When the path returns to the x-axis, the process is complete.

Example 3.1. If D is the Dyck path in Figure 3.1, then we obtain:

P_{D}(1, 1) _{(0, 0) ; (2, 0) ; (4, 0) ; (6, 0) ; (8, 0) ; (10, 0) ; (12, 0) ; 2 3 5 21 43 321 32 4 ; 1}

P_{D}(1, 2) _{(0, 0) ; (2, 0) ; (4, 0) ; (6, 0) ; (8, 0) ; (10, 0) ; (12, 0) ; 2 32 43 5 321 4 ; 1 3 21 ; 21}

P_{D}(1, 3) _{(0, 0) ; (2, 0) ; (4, 0) ; (6, 0) ; (8, 0) ; (10, 0) ; (12, 0) ; 2 3 32 4 ; 1 21 5 21 43 ; 321}

P_{D}(3, 3) _{(0, 0) ; (2, 0) ; (4, 0) ; (6, 0) ; (8, 0) ; (10, 0) ; (12, 0) ; 3 ; 1 21 5 21 43 321 2 32 4}

P_{D}(3, 4) _{(0, 0) ; (2, 0) ; (4, 0) ; (6, 0) ; (8, 0) ; (10, 0) ; (12, 0) ; 2 1 21 32 321 4 ; 43 ; 3 5}

P_{D}(5, 5) _{(0, 0) ; (2, 0) ; (4, 0) ; (6, 0) ; (8, 0) ; (10, 0) ; (12, 0) ; 5 ; 2 32 4 1 21 3 21 43 321}

Now we define the number p_{D}(i, j) by

(3.1)  p_{D}(i, j) := (# of steps in the ascents of P_{D}(i, j)) − 1.

Then we observe

p_{D}(i, j) = (# of blocks in the ascents)

= (# of peaks) + (height of the first peak) − 2.

Recall that we have the map Ψ from the set of Dyck paths into the set of fully commutative elements. Now we state the main result of this section.

Proposition 3.2. Assume that a Dyck path D does not have an ascent longer than 1 step except for an ascent beginning on the x-axis. Then the dimension d_{D} of the homogeneous module S(Ψ(D)) is given by the formula

(3.2)  d_{D} = ∏ \frac{k!}{p_{D}(i, j)}

where k = (sum of peak heights) − (# of peaks) for the path D and the product runs over all blocks T_{i}^{j} on the extended ascents of D.

A proof of the above proposition will be given in the rest of this section. Let us see an example before we begin the proof.

Example 3.3. For the path D in Figure 3.1, the numbers p_{D}(i, j) are shown here:

\begin{matrix}(i, j) & (1, 1) & (1, 2) & (1, 3) & (3, 3) & (3, 4) & (5, 5) \\ p_{D}(i, j) & 1 & 3 & 5 & 1 & 3 & 1 \end{matrix}

Then the dimension of the homogeneous module corresponding to the fully commutative element Ψ(D) = 321435 is

d_{D} = \frac{6!}{1 · 3 · 5 · 1 · 3 · 1} = 16.

Recall that an element w ∈ S_{n+1} is called dominant minuscule if there is a dominant integral weight Λ and a reduced expression w = s_{i_{1}}s_{i_{2}} ⋯ s_{i_{d}} such that

s_{i_{k}}s_{i_{k+1}} ⋯ s_{i_{d}}Λ = Λ − α_{i_{k}} − α_{i_{k+1}} − ⋯ − α_{i_{d}}, 1 ≤ k ≤ d.

It is known that dominant minuscule elements are fully commutative. We have the following characterization of dominant minuscule elements.

Proposition 3.4 ([19, Proposition 2.5]). If w = s_{i_{1}}s_{i_{2}} ⋯ s_{i_{d}} ∈ S_{n+1} is a reduced expression, then w is dominant minuscule if and only if the following two conditions are satisfied:

(1) between every pair of occurrences of a generator s_{i} (with no other occurrences of s_{i} in between) there are exactly two generators (possibly equal to each other) that do not commute with s_{i};

(2) the last occurrence of each generator s_{i} is followed by at most one generator that does not commute with s_{i}.

For a Dyck path D, it is clear that Ψ(D)^{−1} is also a fully commutative element. The following corollary characterizes dominant minuscule elements using shapes of Dyck paths. One can compare it with Lemma 3.9 in [12], where straight shapes are used.

Lemma 3.5. Let D be a Dyck path. Then Ψ(D)^{−1} is dominant minuscule if and only if any ascent in D not beginning on the x-axis has a length of 1.

Proof. Assume that D has no ascents longer than 1 besides those that begin on the xaxis. Since we must have a descent and then an ascent to get from one peak to the next, we see that the condition (i) of Proposition 3.4 is satisfied by Ψ(D) and Ψ(D)^{−1}. Write Ψ(D) = T_{i_{1}}^{m_{1}}T_{i_{2}}^{m_{2}} ⋯ T_{i_{ℓ}}^{m_{ℓ}} as before. Then every generator in T_{i_{1}}^{m_{1}} first appears in Ψ(D) and there is at most one generator before its occurrence that does not commute with it. Thus Ψ(D)^{−1} satisfies the condition (ii) of Proposition 3.4 with the generators in T_{i_{1}}^{m_{1}}.

Consider now the peak corresponding to the segment T_{i_{2}}^{m_{2}}. If we arrive there after an ascent of length 1, then m_{2} = m_{1} + 1 and i_{1} < i_{2}. Thus the only new generator appearing in T_{i_{2}}^{m_{2}} is s_{m_{2}} = s_{m_{1}+1} and it commutes with all the generators preceding it except s_{m_{1}}. On the other hand, if we arrive at this peak after following an ascent longer than 1 step then, by assumption, this ascent begins on the x-axis. Then, we necessarily find that i_{2} > m_{1}+1. So every generator in T_{i_{2}}^{m_{2}} appears here for the first time, but commutes with all generators appearing previously. We can continue inductively, analyzing the generators appearing for the first time in each segment T_{i_{j}}^{m_{j}}, and see that the condition (ii) of Proposition 3.4 is satisfied by Ψ(D)^{−1}. Therefore, the element Ψ(D)^{−1} is dominant minuscule.

Conversely, if Ψ(D)^{−1} is dominant minuscule, the condition (ii) of Proposition 3.4 implies that any generator appearing for the first time in a segment T_{i_{j}}^{m_{j}} will either commute with all previously appearing generators (thus the ascent begins on the x-axis), or that it does not commute with exactly one previously appearing generator (thus m_{j−1} = m_{j} − 1, and the ascent is of length 1). □

Proof of Proposition 3.2. We will obtain the formula (3.2) as a reformulation of the Peterson–Proctor formula [12, Theorem 3.10]. Write w = Ψ(D). It follows from Lemma 3.5 that w^{−1} is dominant minuscule. Then we only need to establish two things: first, a bijective correspondence between

{β ∈ Δ^{+} : w(β) < 0} and {P_{D}(i, j) : T_{i}^{j} is on the extended ascents of D},

where Δ^{+} is the set of positive roots; second, the equality ht(β) = p_{D}(i, j) when β corresponds to the path P_{D}(i, j).

We write Ψ(D) = T_{i_{1}}^{m_{1}}T_{i_{2}}^{m_{2}} ⋯ T_{i_{ℓ}}^{m_{ℓ}}. Each β ∈ Δ^{+} with w(β) < 0 determines a unique (i_{k}, n_{k}), i_{k} ≤ n_{k} ≤ m_{k}, such that

β = α_{n_{k}}T_{i_{k}}^{n_{k}−1}T_{i_{k+1}}^{m_{k+1}} ⋯ T_{i_{l}}^{m_{l}} = α_{n_{k}}T_{i_{k}}^{n_{k}−1}T_{i_{k+1}}^{n_{k}+1}T_{i_{k+2}}^{n_{k}+2} ⋯ T_{i_{l}}^{n_{k}+l−k},

where the action on α_{n_{k}} is from the right. On the other hand, each block T_{i_{k}}^{n_{k}}, i_{k} ≤ n_{k} ≤ m_{k}, is on an extended ascent and

Ψ(P_{D}(i_{k}, n_{k})) = T_{i_{k}}^{n_{k}}T_{i_{k+1}}^{n_{k}+1}T_{i_{k+2}}^{n_{k}+2} ⋯ T_{i_{l}}^{n_{k}+l−k}.

Then the correspondence β ↦ P_{D}(i_{k}, n_{k}) is clearly one-to-one and onto.

Furthermore, we see that

β = α_{n_{k}}T_{i_{k}}^{n_{k}−1}T_{i_{k+1}}^{n_{k}+1}T_{i_{k+2}}^{n_{k}+2} ⋯ T_{i_{l}}^{n_{k}+l−k} = (α_{i_{k}} + ⋯ + α_{n_{k}}) + α_{n_{k}+1} + ⋯ + α_{n_{k}+l−k},

and ht(β) = n_{k} − i_{k} + 1 + l − k = (# of steps in the ascents) − 1 = p_{D}(i_{k}, n_{k}) from (3.1). This completes the proof. □

Even when Proposition 3.2 does not apply directly, we may still find the dimension of the corresponding module: we can

• consider the reverse path (reflected left to right), or

• invert the corresponding fully commutative element, and consider the associated Dyck path.

Note that reversing a path corresponds to the graph automorphism of the Dynkin diagram. The two options would give distinct paths, but if either satisfies the condition of Proposition 3.2, then we can obtain the correct dimension using the formula.

Example 3.6. The path D in Figure 3.2 below does not satisfy the condition of Proposition 3.2. However, we note that the reverse of the path D is nothing but the path in Figure 3.1, for which we computed the dimension in Example 3.3. Thus we obtain the same dimension, 16, for the homogeneous representation corresponding to D.

Figure 3.2. A Dyck Path for which the formula does not work directly

References

1. Georgia Benkart, Seok-Jin Kang, Se-jin Oh, and Euiyong Park, Construction of irreducible representations over Khovanov–Lauda–Rouquier algebras of finite classical type, Int. Math. Res. Notices 2014 (2014), no. 5, 1312–1366.

2. Riccardo Biagioli, Frédéric Jouhet, and Philippe Nadeau, Fully commutative elements in finite and affine Coxeter groups, Monatsh. Math. 178 (2015), no. 1, 1–37.

3. Leonid A. Bokut and Long-Sheng Shiao, Gröbner–Shirshov bases for Coxeter groups, Comm. Algebra 29 (2001), no. 9, 4305–4319, Special issue dedicated to Alexei Ivanovich Kostrikin.

4. Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov– Lauda algebras, Invent. Math. 178 (2009), no. 3, 451–484.

5. Emeric Deutsch, Dyck path enumeration, Discrete Math. 204 (1999), no. 1–3, 167–202.

6. , Sequence A140717, The On-Line Encyclopedia of Integer Sequences, 2008, published electronically at http://oeis.org.

7. C. Kenneth Fan, A Hecke algebra quotient and some combinatorial applications, J. Algebraic Combin. 5 (1996), no. 3, 175–189.

8. David Hill, George Melvin, and Damien Mondragon, Representations of quiver Hecke algebras via Lyndon bases, J. Pure Appl. Algebra 216 (2012), no. 5, 1052–1079.

9. Vaughan F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388.

10. Seok-Jin Kang and Masaki Kashiwara, Categorification of highest weight modules via Khovanov– Lauda–Rouquier algebras, Invent. Math. 190 (2012), no. 3, 699–742.

11. Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups I, Represent. Theory 13 (2009), no. 09, 309–347.

12. Alexander Kleshchev and Arun Ram, Homogeneous representations of Khovanov–Lauda algebras, J. Eur. Math. Soc. 12 (2010), no. 5, 1293–1306.

13. , Representations of Khovanov–Lauda–Rouquier algebras and combinatorics of Lyndon words, Math. Ann. 349 (2011), no. 4, 943–975.

14. Aaron D. Lauda and Monica Vazirani, Crystals from categorified quantum groups, Adv. Math. 228 (2011), no. 2, 803–861.

15. George Lusztig, Introduction to quantum groups, Springer, 2011.

16. Raphaël Rouquier, 2-Kac–Moody algebras, (2008), arXiv:0812.5023 [math.RT].

17. John R. Stembridge, On the fully commutative elements of Coxeter groups, J. Algebraic Combin. 7 (1996), 353–385.

18. , The enumeration of fully commutative elements of Coxeter groups, J. Algebraic Combin. 7 (1998), no. 3, 291–320.

19. , Minuscule elements of Weyl groups, J. Algebra (2001), 722–743.

Department of Mathematics and Statistics, Haverford College, Haverford, PA 19041, U.S.A.

E-mail address: gfeinberg@haverford.edu

Department of Mathematics, University of Connecticut, Storrs, CT 06269, U.S.A.

E-mail address: khlee@math.uconn.edu