On Partitions and Arf Semigroups

Nesrin Tutaş

doi:10.1515/math-2019-0025

Article Open Access

On Partitions and Arf Semigroups

Nesrin Tutaş

Published/Copyright: May 11, 2019

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

In this study we examine some combinatorial properties of the Arf semigroup. In previous work, the author and Karakaş, Gümüşbaş defined an Arf partition of a positive integer n. Here, we continue this work and give new results on Arf partitions. In particular, we analyze the relation among an Arf partition, its Young dual diagram, and the corresponding rational Young diagram. Additionally, this study contains some results that present the relations between partitions and Arf semigroup polynomials.

Keywords: Arf numerical semigroup; Young diagram; partition; Arf partition

MSC 2010: 20M14; 05A17; 11D07

1 Introduction

A partition λ = [λ₁, λ₂, …, λ_r] of a positive integer n is a non-increasing list of positive integers, λ_r ≤ λ_r−1 ≤ ⋯ ≤ λ₁, whose sum is n and length is r. If λ_i ≠ λ_i+1, 1 ≤ i ≤ r − 1, then λ is called a strict dominant partition.

Partitions occur in several branches of physics and mathematics such as representation theory and coding theory, see [12, 13]. Partitions can be visualized with Young diagrams, see [9, 17]. The Young diagram of a partition λ consists of a left-justified shape of r columns of boxes with lengths λ₁, λ₂, …, λ_r. Flipping a Young diagram over its main diagonal (from upper left to lower right) gives the conjugate diagram. The conjugate partition of λ is the partition corresponding to the conjugate diagram of the Young diagram of λ.

For example, we consider the Young diagram of the partition λ = [4, 3, 1]. In the Young diagram of λ, we have 4 boxes in the first column, we have 3 boxes in the second column and one box in the third column. Hence, we obtain Young diagrams of λ and the conjugate partition of λ, respectively, as follows:

In a Young diagram, the number of boxes in a column (or a row) is called the length of that column (or, respectively, that row). The length of a row is at most the number of columns of the diagram, and there may be more than one row with the same length.

Assume that there are r columns in a Young diagram and there are u_i rows of length i, for each i = 1, 2, …, r, u_i ≥ 0. Then we denote such a Young diagram of the form (shape) Y = 1^u₁2^u₂3^u₃ ⋯ r^u_r and we have n=∑j=1rjuj . If there is no row of length j, 1 ≤ j ≤ r, then u_j = 0 and we omit j⁰ in the presentation of a Young diagram Y.

If λ = [λ₁, λ₂, …, λ_r] is the partition corresponding to Y = 1^u₁2^u₂ ⋯ r^u_r, then

λj=∑i=jrui,1≤j≤r.

Note that

λj−λj+1=uj for each j=1,…,r−1 and λr=ur.

For example, if λ = [4, 3, 3, 2], then Y = 1¹3¹4² and we have the following diagram;

The correspondence λ → Y_λ is a bijection between the set of partitions of positive integers and the set of Young diagrams.

The Young tableau (plural, “tableaux”) of a Young diagram is obtained by placing the numbers 1, …, m in the m boxes of the diagram. A standard Young tableau is a Young tableau in which the numbers form an increasing sequence along each line and along each column.

Given a box of a diagram, the shape formed by the boxes directly to the right of it, the boxes directly below it and the box itself is called the hook of that box. The number of boxes in the hook of a box is called the hook length of that box. Thus a Young diagram with the hook lengths of boxes in it is a Young tableau. Thus we can identify the Young diagram of a partition with a Young tableau.

The hook set of a partition λ is the set of hook lengths of the Young diagram of λ. The hook set of a partition encodes information about the other combinatorial objects related to that partition. The most famous is the hook-length formula which gives the degree of the corresponding irreducible representation of the symmetric group and also counts the number of standard Young tableaux that have the shape of that partition, see [9, 13].

We denote the set of positive integers by ℕ and we put ℕ₀ = ℕ ∪ {0}. The cardinality of any set K will be denoted by |K|. For two subsets A, B of ℕ₀, we set

A+B={u+v:u∈A,v∈B},kA=A+A+⋯+A.

A numerical semigroup S is a monoid of ℕ₀, and it has a finite complement G(S) := ℕ₀ ∖ S. The elements of G(S) are called gaps of S and g := |G(S)| is called the genus of S. The largest element of G(S) is called the Frobenius number and denoted by F(S). The conductor of S is c := F(S) + 1. We say that S is generated by X ⊆ S, if S={∑i=1mhixi:m,hi∈N0, xi∈X, 1≤i≤m} . In this case, X is a system of generators of S and we denote S by 〈X〉. If X = {x₁, …, x_k}, then we write S = 〈x₁, …, x_k〉. Note that a system of generators of a numerical semigroup is a minimal system of generators if none of its proper subsets generates the numerical semigroup. Let {n₁ < n₂ < ⋯ < n_e} be the minimal system of generators of S. Then n₁ is known as multiplicity, and e is called the embedding dimension of S.

If S is a numerical semigroup, then we assume S = {0 = s₀, s₁, …, s_r, ⟶}, where “⟶” means that all subsequent natural numbers which are bigger then s_r belong to S and r denotes the number of small elements of S.

Numerical semigroups have several applications to branches of mathematics such as algebraic geometry, number theory, coding theory. For example, the computation of the minimum distance of algebraic geometric codes involves computations in the Weierstrass semigroup, see [5].

A connection between Young diagrams and numerical semigroups was extended by [7, 15]. A partition λ with no hook lengths divisible by a is called an a-core partition. The set of a-cores is infinite but the number of partitions that are both a-cores and b-cores, simultaneous(a, b)-cores, is finite. In [7], the authors studied correspondences between numerical sets (subsets of ℕ₀ which have finite complement and contain zero) and partitions of a positive integer. They count the set of simultaneous (a, b)-cores that come from semigroups for a certain pair (a, b). Moreover, some formulas for the number of partitions with a given hook set and some asymptotic results for the number of semigroups are given in [7]. In [4], the authors proved that a numerical semigroup is presented by a unique Dyck path of order given by its genus, and analyzed some properties such as weight, symmetry by means of a square diagram.

Given a numerical semigroup S ≠ ℕ₀, we construct a uniquely determined Young diagram and thus a uniquely determined partition as follows. We use the first quadrant of the cartesian xy-plane for the construction by drawing a continuous polygonal path which starts from the origin. Starting with x = 0.

If x ∈ S, then we draw a line segment of unit length to the right.
If x ∉ S, then we draw a line segment of unit length up.
Repeat for x + 1.

For any x greater than the Frobenius number of S we draw a line to the right. The lattice lying above the path and below the horizontal line defines the Young diagram of S. If the Young diagram of a partition λ and a numerical semigroup S are the same, we say that λ is the partition of S. For S = {0, 3, 6, →} and G(S) = {1, 2, 4, 5}, we obtain λ = [4, 2] and we have the following path.

The association of a Dyck path to a numerical semigroup follows from the association of numerical semigroups with sequences of 0s and 1s and then assigned either up-or-right moves to each, for detail see [17]. There are other papers associating paths in the plane to numerical semigroups and vise versa, for instance [11].

Unless otherwise stated we will make the following assumptions and notations:

λ = [λ₁, λ₂, …, λ_r], r∈N, ∑i=1rλi=n and λ1r:=[λ1,…,λ1⏟r times] .
Y_S : The Young diagram corresponding to a numerical semigroup S.
The jth column of Y_S is denoted by G_j, for each j ≥ 0. The set of hook lengths of boxes which are in the jth column is identified with G_j. The construction of Y_S implies that the jth column G_j corresponds to s_j, j = 0, 1, … r − 1. For j ≥ r, s_j is greater than or equal to the Frobenius number of S, then there is no box for s_r at the diagram. We know that G₀ = G(S). Given a box in the G₀, the number of the boxes below shows the number of gaps before the hook length in a given box.

Here note that λ = [2, 2] can not represent a numerical semigroup. Otherwise, 1 ∈ S and S must be ℕ₀, but 2, 3 are not in S = {0, 1, 4, →}. We have the following tableau

Thus the correspondence λ ⟶ S is not a bijection between the set of partitions and the set of numerical semigroups. However, the correspondence S⟶Y_S is a bijection between the set of numerical sets and the set of Young diagrams.

Numerical semigroups have become important because of their applications in algebraic geometry. Valuations of analytically unramified one-dimensional local Noetherian domains are numerical semigroups under certain conditions, and many properties of these rings can be characterized in terms of their associated numerical semigroups, see [2, 3, 18]. Du Val showed geometrically how multiplicity sequences of the blow-ups of a curve can be used to classify singularities. Arf showed the algebraic counterpart of Du Val’s results. Arf’s aim was to calculate the Arf ring closure of the coordinate ring of a curve and then its value semigroup (which is an Arf numerical semigroup), see [2, 10].

A numerical semigroup S is called an Arf semigroup if x + y − z ∈ S, for all x, y, z ∈ S with z ≤ y ≤ x. This property is equivalent to 2x − y ∈ S, for all x, y ∈ S with y ≤ x. For example, ℕ₀ and S = {0, 7, 14, 21, 24, 27 →} are Arf numerical semigroups. The Arf closure of a numerical semigroup S is the smallest (with respect to set inclusion) Arf semigroup containing S. There are several equivalent conditions on Arf semigroups, see [3, 8, 10, 14, 18]. In [10, 14], the authors give parametrizations of numerical semigroups with multiplicity up to 5. In [19], an algorithm is given for finding the Arf closure of a numerical set.

Here, we investigate the properties of the Arf partitions of a positive integer using the set of gaps of an Arf numerical semigroup. In Section 2, firstly, we explain the connection between Arf semigroup and Young tableau. Also, we determine partitions of some numerical semigroup families; Proposition 2.7 is about the partitions of Lipman semigroups (see page 347). Let S_(k) be a numerical semigroup with the minimal system of generators < 4, k, k + t, k + t + 2 >, where k ≡ 2(mod 4) and t is an odd integer with t ≥ 7. Then, we obtain the Arf partition of S_(k), more precisely, in Proposition 2.8 we show that this partition is

3k+2t−44,3k+2t−44−3,…,3+t+12,t+12,t−12,…,2,1.

The intersection of two semigroups gives a binary operation over a subset of the set of partitions of positive integers. We denote this operation by ⨁ and it is detailed in the proof of Theorem 2.9, and we prove that the set of partitions obtained from the sets of gaps of all numerical semigroups is a semigroup with the operation ⨁. In particular, we obtain that the set of Arf partitions is a semigroup with the operation ⨁.

Let λ = [λ₁, …, λ_r] ∈ ℕ^r be a partition of length r. If β_i = λ₁ − λ_r+1−i, 1 ≤ i ≤ r, then β is called the dual partition of λ. For more details on the concept of the duality, see [9]. In Section 3, we define the Young dual of a numerical semigroup using the concept of the dual partition. Given a numerical semigroup S, we determine the elements of the numerical set D which is the Young dual of S, and we give conditions for D to be a numerical semigroup and an Arf semigroup, see Propositions 3.4 and 3.5. Let λ be an Arf partition of a natural number n, and r be the length of λ. Then we show that the dual of λ is also a partition of n with the same length, but it may not be an Arf partition. Furthermore, for the rational diagram of a partition λ, defined in Definition 2, we analyze the behavior of the numerical semigroup corresponding to λ. Corollary 3.7 states that for any Arf partition λ, there exists a partition β such that the rational diagram of β can be represented with λ as denominator and another Arf partition.

In Section 4, we give some relations between semigroup polynomials and Arf partitions (Lemma 4.1 and Theorem 4.2), and we achieve the generating functions of semigroups given in Proposition 2.7 and Proposition 2.8.

2 Arf Semigroup and Young Tableau

Let S be a numerical semigroup of genus g and G(S) = {b₁, …, b_g}. We set α (S) = (α₁, … α_g) with α_i = b_i − i, for all i ≤ g, which is called the Schubert index of S. The sum w(S)=∑i=1gαi is said to be the weight of S. The notion of the weight w(S) indicates the difference between the semigroup < g + 1, …, 2g + 1 > and S.

Lemma 2.1

Let S be a numerical semigroup and λ = [λ₁, …, λ_r] be the corresponding partition. Then the Schubert index of S is determined by the conjugate partition of [λ₂, …, λ_r] and w(S)=∑i=2rλi .

Proof

The proof follows from definitions.□

For example, if S = {0, 3, 6, 8, ⟶}, we have the following Young tableau

and the corresponding partition is λ = [5, 3, 1], α (S) = (0, 0, 1, 1, 2) and w(S) = 4. Note that [3, 1] and the reverse ordering of [1, 1, 2] are conjugate.

For a given numerical semigroup S, we have several related semigroups. For each i ≥ 0, S_i and S(i) are defined as follows:

Si={s∈S:s≥si}S−si={s−si∈N0:s∈S}S(i)=S−Si={z∈N0:z+Si⊆S}.

It is obvious that every S(i) is a numerical semigroup, and we obtain a semigroup chain:

⋯⊂Sr⊂Sr−1⊂⋯⊂S1⊂S=S(0)⊂S(1)⊂⋯⊂S(r)=N0.

For 1 ≤ i ≤ r, we define ith type set T(i) := S(i) ∖ S(i − 1) and t_i := |T(i)| . We call {ti}i=1r the type sequence of S. The Lipman semigroup of S is defined by L(S) = ⋃_k≥1(kS₁ − kS₁). We have another finite chain of semigroups obtained by Lipman semigroup of S : S = L₀ ⊆ L₁ ⊆ L₂ ⊆ ⋯ ⊆ L_i ⊆ ⋯ where L_i(S) := L(L_i−1(S)) is the ith Lipman semigroup of S.

Theorem 2.2 explains that the behavior of the semigroup S over the Young diagram.

Theorem 2.2

Let S = {0 = s₀, s₁, …, s_r, ⟶} be a numerical semigroup, Y_S be the Young diagram of S, and G_i be the hook set of the ith column of Y_S, S(i) = S − S_i, for 0 ≤ i ≤ r. Let T(i) be the ith type set of S, 1 ≤ i ≤ r. Then the following statements hold:

G_i = ℕ₀ ∖ {s − s_i : s ∈ S, s ≥ s_i} = G₀ − s_i and |G_i| = s_r − r − (s_i − i), 0 ≤ i < r. Moreover, G_i does not contain any element of S, 0 ≤ i < r.
The first hook length of G_i is min {b ∈ G₀ : b > s_i} − s_i, 1 ≤ i < r, the last hook length is F(S) − s_i.
S(i) = ⋂_j≥i (S − s_j) = N0∖⋃j=ir−1Gj .
x ∈ T(i) if and only if x ∈ G_i−1 and x ∉ G_j, i − 1 < j < r.
x ∈ T(i) if and only if i = max{j + 1 : x ∈ G₀ − s_j, j < r}.

Proof

We have S(i) = {z ∈ ℕ₀ : z ∈ S − s_j, j ≥ i} = ⋂_j≥i(S − s_j). Then (1), (2) and (3) are clear by using the construction of the diagram Y_S. (4) Since the ith type set is T(i) = S(i) ∖ S(i − ), we obtain

x∈T(i)⇔x+s∈S, for alls∈S, s>si−1 andx+si−1∉S.⇔x∈Gi−1 and x∉Gj,i−1<j<r.

(5) follows from (4).□

Corollary 2.3

Let S be a numerical semigroup, let λ be the corresponding partition of length r and n_i := |{s : s ∉ ⋃_j≥i G_j, s ≤ F(S) − s_i}|, for 0 ≤ i ≤ r. Then we have n_i−1 − n_i = λ_i − λ_i+1 + (1 − t_i), for 1 ≤ i ≤ r.

Proof

The proof follows from the definition of the type sequence, Theorem 2.2 and the construction of the diagram Y_S.□

Corollary 2.4

Let S be a numerical semigroup of genus g, and λ = [λ₁, …, λ_r] be the corresponding partition of length r. Then the following statements hold:

S is an Arf semigroup if and only if t_i = λ_i − λ_i+1, 1 ≤ i < r, t_r = λ_r.
If S is an Arf semigroup, then λi=g−∑j=1i−1tj , 1 ≤ i ≤ r, where t_j is the jth type of S.

Proof

S is an Arf semigroup if and only if S(i) = S − S_i. Using Corollary 2.3 and Theorem 2.2 (3), we have n_i + λ_i = F(S) − s_i, for i ≤ r. Therefore, t_i = s_i − s_i−1 − 1 = λ_i − λ_i+1.
g = λ₁, λ₂ = g − (s₁ − s₀ − 1) and inductively we have λ_i = g − ∑j=1i−1(sj−sj−1−1) ), i ≤ r.□

Proposition 2.5

If S is an Arf semigroup, then the following statements hold:

If Y_S = 1^u₁2^u₂ ⋯ r^u_r, then u_i ≠ 0, for 1 ≤ i ≤ r.
If Y_S = [λ₁, …, λ_r], then λ_i ≠ λ_i+1, 1 ≤ i < r.

Proof

If S is an Arf semigroup, then g_i − g_i−1 ≤ 2 ≤ F(S), where g_i, g_i−1 ∈ G₀ (equivalently, s_i+1 − s_i ≥ 2, 1 ≤ i ≤ c − r, where c is the conductor of S, s_i, s_i−1 ∈ S). In fact, if g_j − g_j−1 > 2, for some g_j < F(S), then g_j − 1, g_j − 2 ∈ S and 2(g_j − 1) − (g_j − 2) = g_j ∈ G₀. But this is a contradiction. Since u_i = s_i − s_i−1 − 1, λi=∑j=iruj , we obtain u_i ≥ 1, 1 ≤ i ≤ r and λ_i ≠ λ_i+1.□

Lemma 2.6

Let S be a semigroup and G_i be the hook set of the ith column of Y_S, for 0 ≤ i ≤ r, and S(i) = {z ∈ ℕ₀ : z + S_i ⊆ S}. Then S is an Arf semigroup if and only if G_i = ℕ₀ ∖ S(i), and S(i) is Arf, 0 ≤ i ≤ r.

Proof

Using Theorem 2.2, we obtain that the hook set G_i is a subset of the complement of the semigroup S(i), for 0 ≤ i ≤ r. For an Arf semigroup S, we have the following equivalent conditions:

S Arf ⟺S(i)=Si−si=Li(S)⟺S(i)(j)=S(i+j), 1≤i+j≤r

where L_i(S) = L(L_i−1(S)) is the ith Lipman semigroup of S. Hence, G_i = G₀ − s_i = ℕ₀ ∖ S_i − s_i = ℕ₀ ∖ S(i) and G_i+j = G₀ − s_i+j = ℕ₀ ∖ S_i+j − s_i+j = ℕ₀ ∖ S(i + j), 0 ≤ i ≤ r. Thus S(i) is also Arf.□

Hence, the related semigroups with an Arf semigroup S can be obtained over the Young diagram Y_S.

Definition 1

Let λ be a partition of positive integer n. If there exists an Arf semigroup such that the gap set G(S) is the set of hook lengths of the first column of the Young diagram of λ, then λ is called an Arf partition of n.

For any positive integer n has at least one Arf partition λ = [n] and S = {0, n + 1, →}. Let take n = 13. All of the Arf partitions of 13 are [13], [9, 4], [9, 3, 1], [10, 3], [10, 2, 1], [11, 2], [12, 1]. Proposition 2.5 states that an Arf partition is a strict dominant partition.

Determining the Arf partitions of positive integers is equivalent to determining Arf semigroups. Partitions of some semigroup families can be found in Proposition 2.7 and Proposition 2.8.

Proposition 2.7

Let S_(k) be a numerical semigroup with the minimal system of generators < m, km + 1, km + 2, …, km + (m − 1) >, where m ≤ 7 is the multiplicity, k ∈ ℕ. Then ith Lipman semigroup is L_i(S_(k)) = < m, km − im + 1, km − im + 2, …, km − im + (m − 1) > and the corresponding partition is

(m−1)(k−i),(m−1)(k−i−1),…,(m−1).

Proof

We prove the proposition by induction on i.□

Proposition 2.8

Let S_(k) be a numerical semigroup with the minimal system of generators < 4, k, k + t, k + t + 2 >, where k ≡ 2(mod 4) and t is an odd integer with t ≥ 7. Then the corresponding Arf partition to S_(k) is

3k+2t−44,3k+2t−44−3,…,3+t+12,t+12,t−12,…,2,1

Proof

Using induction method, we prove that the set of gaps of S_(k) is

G(S(k))={1,2,3,5,6,7,9,…,k−7,k−5,k−4,k−3,k−1,k+1,k+3,…,k+t−2},

and the conductor is k + t − 1. Hence, the jth part of the partition of S_(k) is

λj=3(k−2)4+(t+1)2−3(j−1),1≤j≤k−24,(t+1)2−j+1 ,1+k−24≤j≤(t+1)2−1.

Using Corollary 3.19 in [18] and induction method, we obtain that S_(k) is an Arf semigroup. Hence, λ is an Arf partition.□

We remark that the intersection of two numerical semigroups is again a numerical semigroup. A consequence of the closure of this operation can be seen in Theorem 2.9.

Theorem 2.9

Let P denote the set of partitions obtained from the set of numerical semigroups. Then P is a semigroup.

Proof

Let S and T be numerical semigroups corresponding to partitions λ and β, respectively. Since S ⋂ T = T ⋂ S, we may assume that F(S) ≥ F(T). Now, we use the following notations: λ = [λ₁, …, λ_r] = 1^u₁2^u₂ ⋯ k^u_k, β = [β₁, ⋯, β_f] = 1^v₁ 2^v₂ ⋯ h^v_h, u_i ≥ 0, v_j ≥ 0, 1 ≤ i ≤ k, 1 ≤ j ≤ h. Let M = {s ∈ S ⋂ T : s < F(S)} = {s_i₁, s_i₂, ⋯ s_{i_l}}. The effect of the intersection of two semigroups can be explained as follows: let λ₁ ≥ β₁. If there is an element b = s_d ∈ S ∖ T, then the corresponding column of the diagram of S must be deleted and b=∑i=1dui+d . Since u_d+1 is the number of consecutive gap numbers which are between s_d and s_d+1, the number u_d+1 must be added to u_d. Thus the previous column has u_d+1 + 1 more boxes for gaps which are between s_d and s_d+1. Hence, we obtain u_d+1 + 1 + u_d consecutive gap numbers in S ⋂ T. Denote α_j = |{b ∈ S ∖ T : s_{i_j} < b < s_{i_j+1}}|, j ≤ l, and let p_j denote the number of the consecutive elements of S ⋂ T which are greater than or equal to s_{i_j}. If t_j denotes the length of the jth gap block of S ⋂ T, then we obtain t_j = (t_j−1 + p_j) and this number repeats mj=∑z=ij+1ij+1uz+αj times. If λ₁ < β₁, the proof follows from the similar argument.

Then the intersection of two semigroups gives a binary operation which is denoted by ⨁ in P: α ⨁ β = γ, where γ=t1m1t2m2⋯tlml, tj=(tj−1+pj) and mj=∑z=ij+1ij+1uz+αj , Associativity is clear as a property of intersection, [0] is unit which represents ℕ₀.□

Example 2.10

Let S = {0, 4, 7, 8, 11, 12, 14, 15, 16, 18, →} and T = {0, 3, 6, 7, 9, 10, 12, →}. Then we have S ∩ T = {0, 7, 12, 14, 15, 16, 18, →}. Y_S = 1³2²4²6¹9¹ = [9, 6, 4, 4, 2, 2, 1, 1, 1], Y_T = 1²2²4¹6¹ = [6, 4, 2, 2, 1, 1]. By using the proof of the Theorem 2.9, we obtain the following integers

α1=|{4}|,p1=1,m1=u1+u2+1=3+2+1=6,α2=|{8,11}|,p2=1,m2=u3+u4+u5+2=2+2=4,α3=0,p3=1,m3=u6+0=1+0=1,α4=0,p4=3,m4=u7+u8+u9+0=1+0=1,

and

t1=(0+1)u1+u2+1t4=(3+3)u7+u8+u9+0t2=(1+1)u3+u4+u5+2t5=(4+2)u10+0=0t3=(2+1)u6+0t6=(5+1)u11+0=0.

Hence we get

YS∩T=1u 1+u 2+1(1+1)u 3+u 4+u 5+2(2+1)u 6+0(3+3)u 7+u 8+u 9+0=16243161=[12,6,2,1,1,1].

1322426191⨁12224161=16243161.

Corollary 2.11

Let A be the set of Arf partitions. Then A is a semigroup with the operation ⨁.

Proof

Because of the fact that the intersection of two Arf semigroups is an Arf semigroup, the proof follows from Proposition 2.5 and Theorem 2.9.□

With the same notation as in the proof of Theorem 2.9, we have that α ⨁ β = γ = 1^m₁2^m₂ ⋯ l^m_l, where α, β ∈ A, mj=∑z=1ij+1−ijuij+z+αj and m_j ≠ 0, for 1 ≤ j ≤ l.

3 The Young dual of an Arf semigroup

Definition 2

For a strict dominant partition λ = [λ₁, …, λ_r], the ratio of Young diagrams of the partitions [−v_r+1, −v_r, …, −v_k+1] and [v₁, …, v_k] is called the rational diagram of λ, where v = [λ₁, …, λ_r, 0] − r^r+1 = [v₁, …, v_k, v_k+1, …, v_r+1], the separation in two blocks corresponding to the values v_i ≥ 0, or v_j < 0, 1 ≤ i ≤ k, 1 + k ≤ j ≤ r + 1. The rational diagram of λ is denoted by Y[−vr+1,...,−vk+1]Y[v1,…,vk] .

Schubert calculus uses the Young diagrams for polynomials. The rational diagram is used for the calculation of rational Schubert polynomials, see [1].

Note that the block [−v_r, …, −v_k+1] of the rational diagram which corresponds to the values v_j < 0 gives an inverted diagram. Therefore, we consider the reverse ordering for calculating the hook lengths of the block [−v_r+1, …, −v_k+1].

Example 3.1

If we take λ = [7, 5, 2], then we have v = [7, 5, 2, 0] − [3, 3, 3, 3] = [4, 2, −1, −3]. Thus the rational diagram corresponding to λ is Y[3,1]Y[4,2] ,

where the last tableau is the rational diagram of [7, 5, 2] containing the hook lengths of boxes. Here, G₁ = {1, 2, 4, 5} and G₂ = {1, 2, 4} are the gap sets of two numerical semigroups H₁ = {0, 3, 6, →} and H₂ = {0, 3, 5 →}, respectively.

The concept of the dual of a numerical semigroup has been viewed in [3]. Now, we define a new duality concept for a numerical semigroup. The main motivation comes from Schubert calculus (see duality theorem) and it arises naturally by using partitions.

Definition 3

Let S be a numerical semigroup and λ = [λ₁, …, λ_r] ∈ ℕ^r be the corresponding partition.

The partition [λ₁ − 0, λ₁ − λ_r, …, λ₁ − λ₁] is called the dual partition of S, and denoted by d_λ.
The set of hook lengths of d_λ is called the Young dual of S.

Since S is a numerical semigroup, 1 ∈ G(S) and λ₁ − λ₂ > 1. Note that s_r corresponds to the part λ_r+1. Then we may assume λ_r+1 = 0, we need this for compatibility of the construction of the dual partition of a numerical semigroup.

Example 3.2

Now we consider λ = [4, 1]. The duality relation between Y_[4,1] and Y_[4,3] can be seen by the following diagrams

where the double line determines the rational diagram. Hence, we have two hook sets, G = {1, 2, 3, 5} and T = {1, 3, 4, 5}. We obtain G as the hook set of Y_[4,1] and T as the hook set of the diagram Y_[4,3]. We note that Y_[4,1] and Y_[4,3] are dual to each other. But T is not the set of gaps of a numerical semigroup. The rational diagram corresponding to λ and hook sets can be seen as follows:

Therefore, we have two semigroups {0, 3, 4, 5⟶} and {0, 2, 4, 5⟶}. Both are Arf semigroups.

Proposition 3.3

Let S be a numerical semigroup and λ be the corresponding partition. Let c be the conductor of S and G(S) = {b₁, b₂, …, b_λ₁} (resp., G(S_{d_λ}) = {b̄₁, …, b̄_λ₁). Then the following statements hold:

If s_i (resp., s̄_i) denotes the ith element of S (resp., S_{d_λ}), then we have

c=bi+b¯λ1−i+1, 1≤i≤λ1,c=si+s¯r−i, 1≤i≤r.
c2=∑i=1λ1bi+b¯λ1−i+1+∑i=1rsi+s¯r−i .

Proof

Definition of the dual partition gives (1), using (1) and c = λ₁ + r, we obtain (2).□

In general, the set S_{d_λ} is a numerical set but it may not be a semigroup. For S = {0, 6, 8, →}, the partition of S is λ = [6, 1], then d_λ = [6, 5] and S_{d_λ} = {0, 2, 8, →} is not a semigroup.

Proposition 3.4

Let S be a numerical semigroup and D be its Young dual. If the conductor of S is c = s_r, then the following statements hold:

D=(di)=sr−sr−i, i≤r,sr+i−r,i>r.
For any i, j ≤ r, if there exists k ≤ r such that s_m = s_r−i + s_r−j with m = r + s_r−k or m = r − k, then D is a numerical semigroup.

Proof

is clear by the definition. To prove (2), we consider the following cases: let i, j ≤ r and set d := d_i + d_j. Then we have,

d=sr−(sr−i+sr−j−sr)=sr−(sm−sr).

If m = r + s_r−k, then d = s_r − (s_r + s_r−k − s_r) = s_r − s_r−k ∈ D. If m = r − k, then d = s_r + (s_r - s_r−i − s_r−j) = s_r + (s_r − s_r−k) ∈ D.

Take i ≤ r < j. Then d_i = s_r − s_r−i, d_j = s_r + (j − r) and d=sr+(sr−sr−i)+(j−r)⏟u>0∈D . Define t = r + u, then we get d = d_t = s_r + u = s_r+u. Now, take j, i > r. Then i + j − 2r > 0 and we obtain d = s_r + (i − r) + s_r + (j − r) = 2s_r + (i + j − 2r).□

Proposition 3.5

Let S be a numerical semigroup and let D be the Young dual of S. If 2s_r−i − s_r−j ∈ S, for r ≥ i ≥ j, and D is a numerical semigroup, then D is an Arf semigroup.

Proof

If r ≥ i ≥ j, then s_r−i ≤ s_r−j ≤ s_r. Assume that there exists m > 0 such that 2s_r−i − s_r−j = s_m. Then we have

sr−sm=sr−(2sr−i−sr−j)=2sr+(sr−j−2sr−i−sr)sr−sm=2[sr−sr−i]−[sr−sr−j]=2di−dj.

In this case, if m ≤ r, then m = r − t, t ≤ r and d := 2d_i − d_j = s_r − s_r−t = d_t ∈ D. If m > r, then d = s_r − s_m < 0. If u := s_r−j − 2s_r−i ≥ 0, then s_r + (s_r−j − 2s_r−i) = s_r + u and d = 2[s_r − s_r−i] − [s_r − s_r−j] = s_r+u ∈ D.

If j < r < i, then d = 2[s_r + i − r] − [s_r − s_r−j] = s_r + 2(i − r) + s_r−j > s_r and d ∈ D. If i > j > r, then d = s_r + (i − j) > s_r and d ∈ D.□

Corollary 3.6

Let λ = [λ₁, …, λ_r] be an Arf partition of a positive integer n. Then the following statements hold:

The dual of λ is a partition of n and its length is r.
If v = [λ₁, …, λ_r, 0] − r^r+1, then v_p := [v₁, …, v_k] is an Arf partition, where v_i is the ith part of v, v_i ≥ 0, i ≤ k ≤ r + 1.

Proof

Using Proposition 3.4 and the definition of Arf partition, one can obtain the Corollary 3.6 (1). For (2), it is enough to see that the partition v_p = [v₁, …, v_k] presents a semigroup containing the semigroup of λ.□

Corollary 3.7

Let λ = [λ₁, …, λ_r] be an Arf partition. Then there exists a partition β such that the rational diagram of β can be represented with λ as denominator and another Arf partition.

Proof

Define β = [λ₁, …, λ_r, 0] + [r + 1]^r. Then the corresponding rational diagram to β is Y[r]Yλ . Here Y_λ corresponds to the Arf semigroup of λ and Y_[r] is the diagram of the semigroup {0, r + 1, ⟶} which is also Arf.□

4 Arf semigroup polynomial

For a numerical semigroup S, we have

11−x=∑s≥0xs=∑s∈Sxs+∑s∈N0∖Sxs.

HS(x)=∑s∈Sxs is called the generating function associated to S and P_S(x) = (1 − x) ∑s∈Sxs its semigroup polynomial. Here, H_S(x) is not a polynomial but P_S(x) is. On the other hand, we have that

PS(x)=(1−x)HS(x)=1+(x−1)∑s∈N0∖Sxs.

There are several papers dealing with the polynomial P_S(x), see [6] and [16]. We can associate semigroup polynomials with a partition of a natural number n. For G₀ = {ℕ}₀ ∖ S, we have a partition whose hook set is G₀. For any hook number j which occurs in the first column, we form a polynomial involving a sum of powers x^j. Adding a column to the left of the diagram of λ means the multiplication of the polynomial of λ by x. We can illustrate this association in the following table.

partition	tableau	polynomial
λ = [0]	−
λ = [1]		x
λ = [2]		x + x²
λ = [1, 1]		x² = x.x
λ = [2, 1]		x + x.x²
λ = [3, 1]		x + x² + x.x³
λ = [2, 2, 1]		x[x + x³]

Lemma 4.1

Let Y_S = 1^u₁2^u₂ ⋯ r^u_r be the Young diagram of a given semigroup S and let define S(x)=∑s∈N0∖Sxs . Then we have

S(x)=∑j=1r∑i=1ujxi+sj−1.

Proof

The complement of the hook set of the first column of Y_S is

{0,u1+1,u1+u2+2,u1+u2+u3+3,…,u1+u2+⋯+ur+r,⟶}.

Hence, we obtain

S(x)=∑i=1u1xi+xu1+1∑i=1u2xi+xu1+u2+2∑i=1u3xi+…+xu1+…+ur−1+r−1∑i=1urxi

by rearranging, we have S(x)=∑i=1u1xi+xs1∑i=1u2xi+...+xsr−1∑i=1urxi .□

Theorem 4.2

Let S be an Arf semigroup with type sequence {ti}i=1r , and S(k) = S − S_k, 0 ≤ k ≤ r. If S(x) = ∑s∈N0∖Sxs , then the following statements hold:

S(x)=∑j=1r∑i=1tjxi+sj−1 .
S(0)(x) = S(x) and for k ≥ 1, we have

S(k)(x)=∑i=1tk+1xi+∑i=1r−1−k−1∑j=1tk+i+1xj+sk+i−1−∑i=1ksi.
S(k − 1)(x) = ∑i=1tkxi + x^t_k+1 S(k)(x) for k ≥ 1.
The semigroup polynomial of S is P_S(x) = 1 + (x − 1) S(x).

Proof

Since S is an Arf semigroup, we have t_i = u_i by Corollary 2.4. Using Lemma 4.1, Theorem 2.2 and the definition of the semigroup S(k), we obtain (1)-(4).□

Corollary 4.3 follows from Theorem 4.2.

Corollary 4.3

Let S be an Arf semigroup and c be the conductor of S, S(x) = ∑s∈N0∖Sxs . Then the following statements hold:

If s₁ = 2, then S(x)=∑j=0c2−1x2j+1 .
If S = < 3, c + 1, c + 2 > and c ≡ 0(mod 3), then S(x)=∑j=0c3−1x3j(x+x2) .
If S = < 3, c, c + 2 > and c ≡ 2(mod 3), then S(x)=∑j=0c−23−1x3j(x+x2)+xc−1 .

In Proposition 2.7 and Proposition 2.8, we obtained partitions of some semigroup families. The generating functions associated to these families are given in Corollary 4.4 and Corollary 4.5.

Corollary 4.4

Let S_(k) = < 4, k, k + t, k + t + 2 > and t be an odd integer with t ≥ 7, k ≡ 2(mod 4). Then S(k)(x)=xk+t+xk+t−2+xk−x−x2−x3x4−1 and the generating function associated to S_(k) is

HS(k)(x)=xk+t1−x2+xk+11−x4.

Proof

Let k = 4v + 2, t = 7 + 2a, a ≥ 0. Proposition 2.8 states the partition of S_(k) and we get Y_S = 1³2³ ⋯ v³(v + 1)⋯ (v + a + 4). Since S_(k) is an Arf semigroup, the type sequence is {3,…,3⏟v times,1,…,1⏟a+4 times } . Therefore,

S(k)(x)=(∑l=13xl)(1+x4+x8+…+x4(v−1))+x4v+1(1+x2+x4+…+x2a+6) =x4v−1x3+x2+x+x4v+1x2a+8−1(x2+1)x4−1=−x−x2−x3+xk+t+xk+xk+t+2x4−1.

Hence, P_{S_(k)}(x) = 1 + (x − 1) ∑s∈N0∖S(k)xs=xk+t+xk+xk+t+2+1x+1x2+1 and the generating function associated to S_(k) is

HS(k)(x)=PS(k)(x)(1−x)=xk+t+xk+t+2+xk+11−x4=xk+t1−x2+xk+11−x4.

□

Corollary 4.5

Let S_(k) be a numerical semigroup with the minimal system of generators < m, km + 1, km + 2, …, km + (m − 1) >, and L_i(S_(k)) be the ith Lipman semigroup of S_(k), where m ≤ 7 is the multiplicity of S_(k), 0 ≤ i < k, k ∈ ℕ. Then Li(S(k))(x)=(xmk−i−1)(xm−x)(xm−1)(x−1) , for 0 ≤ i ≤ k − 1, and the generating function associated to L_i(S_(k)) is

HLi(S(k))(x)=x(k−i)m(1−x)−1(xm−1).

Proof

We see that S_(k) is an Arf semigroup and L_i(S_(k)) is also Arf, 0 ≤ i < k. Proposition 2.7 states the partition of the semigroup L_i(S_(k)), and we get Y_{L_i(S_(k))} = 1^m−12^m−1 ⋯ (k − i)^m−1. Using Corollary 2.4, we obtain the type sequence of L_i(S_(k)), {tj}j=1k−i = {m − 1, …, m yq 1}. By Theorem 4.2, we get

Li(S(k))(x)=∑l=1m−1xl+xm∑l=1m−1xl+…+x(k−i−1)m∑l=1m−1xl=(xmk−i−1)(xm−x)(xm−1)(x−1).

Therefore, PLi(S(k))(x)=xmk+1−i−x(k−i)m+1+x−1xm−1, , the generating function associated to L_i(S_(k)) is

HLi(S(k))(x)=x(k−i)m(xm−1)+x−1(xm−1)(1−x)=x(k−i)m(1−x)−1(xm−1).

□

Corollary 4.6

Let S be an Arf semigroup with type sequence {ti}i=1r and D be its Young dual. Then

D(x)=∑j=1r∑i=1tr−j+1xi+sr−sr+1−j.

Proof

If λ = [λ₁, …, λ_r] is the partition of S, then t_i = λ_i − λ_i+1, 1 ≤ i < r and t_r = λ_r. The Young dual of S is D = {0, s_r − s_r−1, s_r − s_r−2, …, s_r − s₁, s_r, ⟶} and the dual partition is d_λ = [λ₁, λ₁ − λ_r, …, λ₁ − λ₂]. Hence, v₁ = λ_r = t_r, v_i = (λ₁ − λ_r−i+2) − (λ₁ − λ_r−i+1) = λ_r−i+1 − λ_r−i+2 = t_r−i+1, 2 ≤ i ≤ r. In other words, the sequence v₁, …, v_r is the reverse ordering of t₁,, …, t_r. By Theorem 4.2, we obtain D(x)=∑j=1r∑i=1tr−j+1xi+sr−sr+1−j .□

Acknowledgements

The author would like to thank the anonymous referee for giving helpful comments.

References

[1] Aker K., Tutaş N., Rational Schubert polynomials, Turk J. Math., 2015, 39, 439-452.10.3906/mat-1409-47Search in Google Scholar

[2] Arf C. Une interprétation algébrique de la suite ordres de multiplicité ďune branche algébrique, Proc. London Math Soc., 1949, 20, 256-287.10.1112/plms/s2-50.4.256Search in Google Scholar

[3] Barucci V., Dobbs D.E., Fontana M., Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains, Mem. Am. Math. Soc., 1997, 125(598), 1-77.10.1090/memo/0598Search in Google Scholar

[4] Bras-Amoros M., Mier A., Representation of numerical semigroups by Dyck paths, Semigroup Forum, 2007, 75, 676-681.10.1007/s00233-007-0717-7Search in Google Scholar

[5] Campillo A., Farran J.I., Munuera C., On the parameters of algebraic-geometry codes related to Arf semigroups, IEEE Trans. Inform. Theory., 2000, 46, 2634-2638.10.1109/18.887872Search in Google Scholar

[6] Ciolan A., Garcia-Sanchez P.A., Moree P., Cyclotomic numerical semigroups, SIAM J. Discrete Math., 2016, 30(2), 650-668.10.1137/140989479Search in Google Scholar

[7] Constantin H., Houston-Edwards B., Kaplan, N., Numerical sets, core partitions, and integer points in polytopes, Combinatorial and Additive Number Theory II– CANT 2015 and 2016, Springer Proc. Math. Stat., Springer, 2017, 220, 99-127.10.1007/978-3-319-68032-3_7Search in Google Scholar

[8] D’anna M., Type sequences of numerical semigroups, Semigroup Forum, 1998, 56, 1-31.10.1007/s00233-002-7001-7Search in Google Scholar

[9] Fulton W., Young Tableaux, With Application to Representation Theory and Geometry, 1997, New York: Cambridge Univ. Press.10.1017/CBO9780511626241Search in Google Scholar

[10] Garcia-Sánchez P.A., Karakaş H..I., Heredia B.A., Rosales JC., Parametrizing Arf numerical semigroups, J. Algebra Appl., 2017, 16(11), 1750209.10.1142/S0219498817502097Search in Google Scholar

[11] Hellus M., Waldi R., On the number of numerical semigroups containing two coprime integers p and q, Semigroup Forum, 2015, 90(3), 833-842.10.1007/s00233-015-9710-8Search in Google Scholar

[12] Huffman W.C., Pless V., Fundamentals of Error-Correcting Codes, 2003, New York: Cambridge Univ. Press.10.1017/CBO9780511807077Search in Google Scholar

[13] James G., Kerber A., The Representation Theory of the Symmetric Groups. Encyclopedia of Mathematics, 16, 1981, Addison-Wesley.Search in Google Scholar

[14] Karakaş. H.I., Parametrizing numerical semigroups with multiplicity 5, Int. J. Algebr. Comput., 2018, 28(01), 69-95.10.1142/S0218196718500042Search in Google Scholar

[15] Keith W.J., Nath R., Partitions with prescribed hooksets, J. Comb. and Num. Thy., 2011, 3(1), 39-50.Search in Google Scholar

[16] Moree P., Numerical Semigroups, cyclotomic polynomials and Bernoulli numbers, Amer. Math. Monthly, 2014, 121, 890-902.10.4169/amer.math.monthly.121.10.890Search in Google Scholar

[17] Olsson J.B., Combinatorics of the representation theory of the symmetric groups Vorlesungen aus dem FB Mathematik der Univ. Essen, Heft 20, 1993.Search in Google Scholar

[18] Rosales J.C., Garcia-Sánchez P.A., Numerical Semigroups, 2009, New York: Springer.10.1007/978-1-4419-0160-6Search in Google Scholar

[19] Tutaş N., Karakaş H.I. and Gümüşba ş N., Young tableaux and Arf partitions, Turk J. Math., 2019, 43, 448-459.10.3906/mat-1807-181Search in Google Scholar

Received: 2018-08-29

Accepted: 2019-02-25

Published Online: 2019-05-11

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0025

Keywords for this article

Arf numerical semigroup; Young diagram; partition; Arf partition

Creative Commons

BY-NC-ND 4.0