Abstract
In this paper, we define the concept of the Study-type determinant, and we present some properties of these determinants. These properties lead to some properties of the Study determinant. The properties of the Study-type determinants are obtained using a commutative diagram. This diagram leads not only to these properties but also to an inequality for the degrees of representations and to an extension of Dedekind’s theorem.
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The quaternions were discovered in 1843 by William Rowan Hamilton. Since then, quaternions have played an important role in mathematics. The Study determinant is known as one of determinants of matrices in quaternions.
In this paper, we define the concept of the Study-type determinant, and we present some properties of these determinants. These properties lead to some properties of the Study determinant.
1. Introduction
Let be a commutative ring, let be a ringFootnote1 that is a free right -module of rank , and let be a left regular representation from to , where is the set of all matrices with elements in . We define the Study-type determinant as
where is the inclusion map from to . The Study-type determinant has the following properties.
For all , the following hold:
(S1) ;
(S2) is invertible in if and only if is invertible in ;
(S3) if is obtained from by adding a left-multiple of a row to another row or a right-multiple of a column to another column, then we have ;
(S4) if there exists a basis of as a -right module satisfying the following conditions:
(i) is invertible in for any ;
(ii) holds for any ,
then, we have , where is the center of .
In the following, we assume that is a free right -module of rank and is a commutative ring. Then, we have the following properties.Footnote2
For all , the following hold:
(S5) , where we regard as ;
(S6) if there exists a basis of as a -right module satisfying the conditions (i) and (ii), then we have .
The Study-type determinant is a generalization of the Study determinant. The Study determinant was defined by Eduard Study (Study, Citation1920). Let be the quaternion field. The Study determinant is defined using a transformation from . It is known that this determinant has the following propertiesFootnote3 (see, e.g., (Aslaksen, Citation1996)).
For all , the following hold:
() ;
() is invertible in if and only if is invertible;
() if is obtained from by adding a left-multiple of a row to another row or a right-multiple of a column to another column, then we have ;
() ;
() .
The above properties can be derived from the properties of the Study-type determinants.
Let and be left regular representations from to and from to , respectively. The following theorem plays an important role in ascertaining the properties of the Study-type determinants.
Theorem 1.1 (see Theorem 4.4 for proof). The following diagram is commutative:
Theorem leads not only to some properties of the Study-type determinant but also to Corollary and Theorem. Let be a basis of as -module, let be a basis of as -module, let , let be the set of independent commuting variables, and let be the general element for , where is the polynomial ring in with coefficients in . For rings and , we denote the set of ring homomorphisms from to by , and we regard any ring homomorphism as such that for any , where is the unit element of . Let . We assume that there exists a commutative ring such that , and and have the direct sums
where and . Then, we have the following corollary and theorem.
Corollary 1.2 (Corollary 5.2). The following holds:
Theorem 1.3 (see Theorem 5.3 for proof). If is an irreducible polynomial over , then we have
where is the degree of the polynomial .
Corollary leads to an extension of Dedekind’s theorem, while Theorem leads to an inequality characterizing the degrees of irreducible representations of finite groups. Let be the group determinant of the finite group , let be a complete set of inequivalent irreducible representations of over , let be independent commuting variables, let be the polynomial ring in with coefficients in , let be the group algebra of over , let , let be the unit element of , and let be the order of . We extend to satisfy , where . The extension of Dedekind’s theorem mentioned above is the following.
Theorem 1.4 (Extension of Dedekind’s theorem, see Theorem 10.3 for proof). Let be a finite group and let be an abelian subgroup of . Then, writing as , we have
The group determinant is the determinant of a matrix with entries in . (It is known that the group determinant determines the group. For the details, see (Formanek & Sibley, Citation1991) and (Mansfield, Citation1992)). Dedekind proved the following theorem concerning the irreducible factorization of the group determinant for any finite abelian group (see, e.g., (Bartel Leendert van der Waerden, Citation1985)).
Theorem 1.5 (Dedekind’s theorem). Let be a finite abelian group. Then, we have
Frobenius proved the following theorem concerning the irreducible factorization of the group determinant for any finite group; thus, he obtained a generalization of Dedekind’s theorem (see, e.g., (Conrad, Citation1998)).
Theorem 1.6 (Frobenius’ theorem). Let be a finite group. Then we have the irreducible factorization
Since Dedekind’s theorem is a special case of Frobenius’ theorem, we call Frobenius’ theorem a generalization of Dedekind’s theorem. On the other hand, Theorem gives the relation on , and Theorem leads to Dedekind’s theorem. Therefore, we call Theorem an extension of Dedekind’s theorem. That is, the “extension” is used to mean “extend to .”
Let be an abelian subgroup of and let be the index of in . The following extension of Dedekind’s theorem, which is different from the theorem due to Frobenius, is given in (Yamaguchi, Citation2017).
Theorem 1.7. Let be a finite abelian group and let be a subgroup of . Then, for every , there exists a homogeneous polynomial such that and
If , we can take for each .
Theorem is a special case of Theorem . Theorems , and lead to the following corollary.
Corollary 1.8 (Corollary 10.5). Let be a finite group and let be an abelian subgroup of . Then, for all , we have
Note that Corollary follows from Frobenius reciprocity, and it is known that if is an abelian normal subgroup of , then divides for all (see, e.g., (Kondo, Citation2011)).
This paper is organized as follows. In Section , we present an action of the symmetric group on the set of square matrices, and we introduce two formulas for determinants of commuting-block matrices. In addition, we recall the definition of the Kronecker product, and we present a permutation using the Euclidean algorithm. This permutation causes the order of the Kronecker product to be reversed. These preparations are useful for proving Theorem . In Section , we recall the definition of regular representations and invertibility preserving maps, and we show that regular representations are invertibility preserving maps. The regular representation is used in defining Study-type determinants. In addition, we formulate a commutative diagram of regular representations. This commutative diagram is also useful for proving Theorem . In Section , we prove Theorem . In Section , we give a corollary concerning the degrees of some representations contained in regular representations. In Section , we define Study-type determinants and elucidate their properties. In addition, we construct a commutative diagram for Study-type determinants. This commutative diagram leads to some properties of Study-type determinants. In Section , we give a Cayley-Hamilton-type theorem for the Study-type determinant under the assumption that there exists a basis of as a -right module satisfying the conditions (i) and (ii). This Cayley–Hamilton type theorem leads to some properties of Study-type determinants. In Section , we obtain two expressions for regular representations under the assumption that the basis of as a -module satisfying the following conditions:
(iii) for any and , ;
(iv) there exists such that ;
(v) for any , there exists such that .
In addition, we characterize the images of regular representations in the case that satisfies the following additional condition:
(vi) for any and , .
This characterization is the following.
Theorem 1.9 (see Theorem 8.4 for proof). Let be the left regular representation from to with respect to . Then we have
In Section , we introduce the Study determinant and its properties, and we derive these properties from the properties of the Study-type determinant. In addition, from Theorem , we obtain the following characterizations of and .
() ;
() .
In the last section, we recall the definition of group determinants, and we give an extension of Dedekind’s theorem and derive an inequality for the degree of irreducible representations of finite groups.
2. Preparation
In this section, we present an action of the symmetric group on the set of square matrices, and we introduce two formulas for determinants of commuting-block matrices. In addition, we recall the definition of the Kronecker product, and we determine a permutation using the Euclidean algorithm. This permutation reverses the order of the Kronecker product. These preparations are useful for proving Theorem .
2.1. Invariance of determinants under an action of the symmetric group
In this subsection, we present an action of the symmetric group on the set of square matrices. This group action does not change the determinants of matrices.
Let be a ring, which is assumed to have a multiplicative unit , let be the set of all matrices with elements in , let , let , and let be the symmetric group on . We express the determinant of from to as
The group acts on as , where . If is commutative, then the group action does not change the determinants of matrices in . In fact, we have
2.2. Determinants of commuting-block matrices
In this subsection, we introduce two formulas for determinants of commuting-block matrices.
Let . The matrix can be written as , where are matrices. The following is a known theorem concerning commuting-block matrices (Ingraham, Citation1937) and (Kovacs, Silver, & Williams, Citation1999).
Theorem 2.1. Let be a commutative ring, and assume that are commutative. Then we have
Let be the identity matrix of size . We have the following lemma.
Lemma 2.2. Let be a ring, let be a commutative ring, let be a subring of , and let be a ring homomorphism from to . In this case, if for all , then holds.
Proof. (The method used here is based on that of the proof of Theorem in (Kovacs et al., Citation1999)). We prove this by induction on . In the case , the statement is obviously true. Then, assuming that the statement is true for , we prove it for .
Let for all and let . Then, we find that the following equation holds:
Therefore, if is invertible, then holds. Next, suppose that is non-invertible. We embed in the polynomial ring , and replace by . Then, because is neither zero nor a zero divisor, we have . Substituting yields the desired result. □
2.3. Kronecker product and a permutation obtained using the Euclidean algorithm
In this subsection, we recall the definition of the Kronecker product, and then we determine a permutation using the Euclidean algorithm. This permutation reverses the order of the Kronecker product.
Let be an matrix and let be an matrix. The Kronecker product is the matrix
Next, we determine a permutation using the Euclidean algorithm. From the Euclidean algorithm, we know that is a bijection map, where and . Thus, . We have the following lemma.
Lemma 2.3. Let be a commutative ring, let , and let . Then we have .
Proof. For any , by the Euclidean algorithm there exist unique integers and such that and . Therefore, we have
On the other hand, we have
The property described by the above lemma is a special case of a property of the Kronecker product (see, e.g., (Henderson & Searle, Citation1981)). We do not explain this general property, because, for our purposes, it is simpler to use Lemma 2.3.
3. On the left regular representation
In this section, we recall the definition of regular representations and invertibility preserving maps, and we show that regular representations are invertibility preserving maps. A regular representation is used in defining Study-type determinants. In addition, we construct a commutative diagram of regular representations. This commutative diagram is also useful for proving Theorem 4.4.
3.1. Definition of the regular representation
In this subsection, we recall the definition of regular representations, and we give three examples.
Let and be rings and let be the center of . Assume that is a free right -module with an ordered basis . In other words, , and is a subring of . Then, for all , there exists a unique such that
Hence, we have . The injective -algebra homomorphism is called the left regular representation from to with respect to .
Let be the field of real numbers, let be the field of complex numbers, and let be the quaternion field. Below, we give three examples of regular representations.
Example 3.1. Let and let . Then . For all , we have
Example 3.2. Let and let . Then . For all , we have
where is the complex conjugate matrix of .
Example 3.3 Let and let be the trivial group. Then, the group algebra is a finite dimension algebra over with basis . For all , we have
3.2. Definition of the invertibility preserving map
In this subsection, we recall the definition of invertibility preserving maps, and we show that regular representations are invertibility preserving maps. Usually, invertibility preserving maps are defined for linear maps (see, e.g., (Brešar & Peter, Citation1998)). However, we do not assume that invertibility preserving maps are linear maps as in (Yamaguchi & Yamaguchi, Citation2019).
The following is the definition of invertibility preserving maps.
Definition 3.4 (Invertibility preserving map). Let and be rings, and let be a map. Assume that for any , the following condition holds: is invertible in if and only if is invertible in . Then we call an “invertibility preserving map.”
We recall that if is a commutative ring, then we do not need to distinguish between left and right inverses for . Because, is an injective algebra homomorphism, and if is the unit element, then is the unit element.
We denote the unit element of as . In terms of the regular representation, we have the following lemma.
Lemma 3.5. If is a commutative ring, then we have the following properties:
(1) the map is invariant under a change of the basis ;
(2) a left regular representation is an invertibility preserving map.
Proof. First, we prove . Let be another left regular representation from to . Then for all , there exists such that . Therefore, we have . Next, we prove . If is invertible in , then we have . Note that because is a multiplicative map, is invertible. Conversely, if is invertible, then is invertible. Because , we can choose . Then, we have . Therefore, we obtain . Hence, is invertible. This completes the proof.□
3.3. Commutative diagram of regular representations
In this subsection, we present a commutative diagram of regular representations.
Let be the set of natural numbers, let be a ring, and let be a free right -module with an ordered basis . Then, we have the direct sum
for any . We write the left regular representation from to with respect to as and that from to with respect to as . In terms of the regular representations, we have the following lemma.
Lemma 3.6. The following diagram is commutative:
Proof. For all , we have
This completes the proof.□
Let and let be the left regular representation from to with respect to . Then, from Lemma 3.6, we have the following corollary.
Corollary 3.7. The following diagram is commutative:
4. A commutative diagram on the regular representations and determinants
In this section, we prove Theorem . This theorem provides a commutative diagram on the regular representations and determinants. From this commutative diagram, we are able to determine the properties of Study-type determinants, presented in Section . In addition, from this commutative diagram, we are able to derive an inequality for the degrees of representations (Section ) and an extension of Dedekind’s theorem (Section ).
Let be the matrix with in the entry and otherwise. First, we prove the following lemma:
Lemma 4.1. For any , we have
Proof. We express as for any . From , we obtain
Therefore, we have . This completes the proof.□
From Lemmas and , we obtain the following lemma.
Lemma 4.2. For any , we have
Next, from Lemmas and , we obtain the following corollary.
Corollary 4.3. Let be a commutative ring, let be a ring that is a free right -module, let be a subring of , and let and be left regular representations from to and from to , respectively. If for all , then for all .
In the following, we assume that and are commutative rings. Then, we have the following theorem.
Theorem 4.4 (Theorem 1.1). Let and be commutative rings, let be a free right -module, and let and be left regular representations from to and from to , respectively. Then the following diagram is commutative:
Proof. Without loss of generality, we can assume that and . Let . Then, is a ring homomorphism and is a commutative ring, from Theorem and Lemma, we have
This completes the proof.□
5. Degrees of some representations contained in regular representations
In this section, we give a corollary regarding the degrees of some representations contained in regular representations.
The following is the definition of the general element.
Definition 5.1 (General element). Let be a finite set and let be a set of independent commuting variables. The general element for defined as
where is the set of polynomials over .
Let and let be the set of independent commuting variables, and we denote as . For rings and , we denote the set of ring homomorphisms from to by , and we regard any ring homomorphism as such that for any , where is the unit element of .
From Theorem , for any , we have
Let be a commutative ring such that . We assume that and have the following direct sums:
where and . Then, we have the following corollary from Theorem 4.4.
Corollary 5.2 (Corollary 1.2). The following hold:
Corollary 5.2 leads to the following theorem.
Theorem 5.3 (Theorem 1.3). If is an irreducible polynomial over , then we have
where is the degree of the polynomial .
Proof. If is an irreducible polynomial over , then we have
This completes the proof.□
6. On the study-type determinant
In this section, we define the Study-type determinant, and we elucidate its properties. The Study-type determinant is a generalization of the Study determinant. In addition, we present a commutative diagram characterizing Study-type determinants. This commutative diagram allows us to determine some properties of the Study-type determinant.
The following is the definition of the Study-type determinant.
Definition 6.1 (Study-type determinant). Let be a commutative ring, let be a ring that is a free right -module, and let be a left regular representation from to . We define the Study-type determinant as
where is the inclusion map from to .
A Study-type determinant is a multiplicative and invertibility preserving map, because determinants and left regular representations are multiplicative and invertibility preserving maps. Thus, we have the following lemma.
Lemma 6.2 ((S1) and (S2)). Study-type determinants possess the following properties:
(1) a Study-type determinant is a multiplicative map;
(2) a Study-type determinant is an invertibility preserving map.
In addition, we have the following lemma.
Lemma 6.3 ((S3)). If is obtained from by adding a left-multiple of a row to another row or a right-multiple of a column to another column, then we have .
Proof. The property (S) can be restated as , where . Then, from Theorem and Lemma , we have
This completes the proof.□
For a ring and , we denote as . From Corollary and Theorem , we obtain the following commutative diagram:
From this, we obtain the following theorem.
Theorem 6.4 ((S5)). Let and be commutative rings, let be a ring that is a free right -module, and let be a free right -module. Then we have
where we regard as .
Lemmas and and Theorem are equal to (S1), (S2), (S3), and (S5) in Section .
7. Characteristic polynomial and Cayley-Hamilton-type theorem for the study-type determinant
In this section, we give a Cayley-Hamilton-type theorem for the Study-type determinant under the assumption that there exists a basis of as -module satisfying certain conditions. This Cayley-Hamilton-type theorem leads to some properties of Study-type determinants.
Let be a left regular representation from to and let be an independent variable. We write as for all , i.e., we express the characteristic polynomial of as . In this section, we assume that has the following properties:
(i) is invertible in for any ;
(ii) holds for any ;
Then, we have the following lemma.
Lemma 7.1. We have . In particular, .
Proof. Without loss of generality, we can assume that . We show that for all . Since is invertible for all , there exists an invertible element such that . Then, from , we obtain . Therefore, we have . Also, because is injective and , we have . Therefore, we have
This completes the proof.□
The following theorem is a Cayley-Hamilton-type theorem.
Theorem 7.2 (Cayley-Hamilton-type theorem). Let be the characteristic polynomial of . Then we have
Proof. Without loss of generality, we can assume that . From the Cayley-Hamilton theorem for commutative rings, we have . Then, because is a -algebra homomorphism, from Lemma , we obtain
Finally, because is injective, we have . This completes the proof.□
From Corollary and Lemma , we obtain the following corollary.
Corollary 7.3 ((S4)). For all , we have .
Next, from Corollaries 4.3 and 7.3, we obtain the following corollary.
Corollary 7.4 ((S6)). For all , we have .
8. Image of a regular representation when the direct sum forms a group
In this section, we obtain two expressions for regular representations and we characterize the image of a regular representation in the case that a basis of as -module satisfies certain conditions.
Let and we define a product of and as
In this section, we assume that satisfies the following conditions:
(iii) for any and , ;
(iv) there exists such that as set;
(v) for any , there exists such that .
It is easy to show that is invertible in and is a group. For a group , we denote the unit element of as . We remark that even if has the above properties, then and are not necessary group isomorphism.
Example 8.1. Let and . Then, and are groups and basis of as a -right module, respectively. However, as group.
Let be the diagonal matrix . To obtain an expression for , we define the indicator function by
We now formulate an expression for regular representations in terms of
Lemma 8.2. Let , where . Then we have
Proof. For all , we have
This completes the proof.⁏
Let be the regular representation of the group . From
We can obtain an another expression for regular representations.
Corollary 8.3. Let , where . Then we have
We add the following assumption:
(vi) for any and , .
Let for all , and we write as for any map and any set . We show that is an image of if and only if commutes with for all .
Theorem 8.4 (Theorem 1.9). We have
Proof. From Corollary 8.3, we have . We show that . For all , there exists and such that where and for all . Also from Corollary 8.3, we have for all . Further, we know that for all , there exists such that and for all . Therefore, we have
This implies . This completes the proof.⁏
From Lemma 4.1 and Theorem 8.4, we obtain the following corollary.
Corollary 8.5. We have
9. On the relationship between the study-type and study determinants
In this section, we introduce the Study determinant and elucidate its properties. We derive these properties from the properties of the Study-type determinant.
First, we recall that any complex matrix can be written uniquely as , where and , and any quaternionic matrix can be written uniquely as , where and . We define and by
respectively. The Study determinant is defined by
for all . Let
Then, the following are known (see, e.g., (Aslaksen, Citation1996)):
the maps and are injective algebra homomorphisms;
;
is invertible in if and only if is invertible;
if is obtained from by adding a left-multiple of a row to another row or a right-multiple of a column to another column, then we have Sdet(a’) = sdet(a);
;
;
;
.
These properties can be derived from results given in Sections 2–8. Let , let , let , let , and let . Then, the basis of as -module and the basis of as -module satisfy the conditions (i)–(vi). From Examples 3.1 and 3.2, we have and , where and are inclusion maps. Therefore, () holds. Also, from Lemma 6.2, () and () hold. By Lemma 6.3, we have (). From Corollary 7.3, () holds. By Corollary 7.4, () holds. Finally, () and () can be derived from Corollary 8.5 and the fact that if and only if .
10. On the relationship to the group determinant
In this section, we recall the definition of group determinants, and we give an extension of Dedekind’s theorem and derive an inequality for the degrees of irreducible representations of finite groups.
First, we recall the definition of the group determinant. Let be a finite group, let be the set of independent commuting variables, let be the polynomial ring in the variables with coefficients in , and let be the order of . The group determinant of is given by , where we apply a numbering to the elements of (for details, see, e.g., (Conrad, Citation1998), (Frobenius, Citation1896a), (Frobenius, Citation1896b), (Frobenius, Citation1903), (Hawkins, Citation1971), (Johnson, Citation1991), (Bartel Leendert van der Waerden, Citation1985), and (Yamaguchi, Citation2017)). It is thus seen that the group determinant is a homogeneous polynomial of degree . In general, the matrix is covariant under a change in the numbering of the elements of . However, the group determinant, , is invariant.
Let be the group algebra of over , let be an abelian subgroup of , let be the index of in , let , let , and let . For a group , we denote the regular representation of the group as . We regard as -algebra homomorphism from to . Then, from Lemma, is equivalent to the left regular representation from to . Therefore, the following commutative diagram holds:
It is easy to show that (see, e.g., (Conrad, Citation1998)). Therefore, we have
We extend to satisfy , where . Frobenius proved the following theorem concerning the factorization of the group determinant (see, e.g., (Conrad, Citation1998)).
Theorem 10.1 (Frobenius’ theorem, Theorem 1.6). Let be a finite group. Then we have the irreducible factorization
Let be a complete set of inequivalent irreducible representations of over . Theorem 10.1 holds from the following theorem (which is treated in detail in (Steinberg, Citation2012)).
Theorem 10.2. Let be a finite group, let , and let be the left regular representation of . Then we have
Therefore, the following theorem is deduced from Corollary 5.2.
Theorem 10.3 (Extension of Dedekinds’ theorem, Theorem 1.4). Let be a finite group and let be an abelian subgroup of . Then, writing as , we have
The following is a special case of Theorem 10.3 (Yamaguchi, Citation2017).
Theorem 10.4 (Theorem 1.7). Let be a finite abelian group and let be a subgroup of . Then, for every , there exists a homogeneous polynomial such that and
If , then we can take for each .
From Theorems 5.3, 10.1 and 10.3, we obtain the following corollary:
Corollary 10.5 (Corollary 1.8). Let be a finite group and let be an abelian subgroup of . Then, for all , we have
Note that Corollary 10.5 follows from Frobenius reciprocity, and it is known that if is an abelian normal subgroup of , then divides for all (see, e.g, (Kondo, Citation2011)).
11. Future work
There are several noncommutative determinants, and some of their relationships are known. For example, the following is known (see, e.g., (Aslaksen, Citation1996), (Kyrchei, Citation2008), and (Kyrchei, Citation2012)):
For any , we have
where and are the determinant of the Moore (see, e.g., (Aslaksen, Citation1996), (Dyson, Citation1972), and (Kyrchei, Citation2012)) and of the Dieudonné ((Artin, Citation2016) and (Dieudonné., Citation1943)), respectively.
In the future, we study relations with other noncommutative determinants, for example, Capelli identities (see, e.g., (Capelli, Citation1890), (Itoh & Umeda, Citation2001), (Umeda, Citation2008), and (Yamaguchi, Citation2018)), the determinant of Chen (Longxuan, Citation1991), quasideterminant (Gel’fand & Retakh, Citation1991), the determinant of Kyrchei (Kyrchei, Citation2008) and (Kyrchei, Citation2012), the determinant of matrices of pseudo-differential operators (Sato & Kashiwara, Citation1975), etc.
Acknowledgments
I am deeply grateful to Professor Hiroyuki Ochiai, who provided the helpful comments and suggestions. Also, I would like to thank my colleagues in the Graduate School of Mathematics at Kyushu University, in particular Yuka Yamaguchi, for comments and suggestions. This work was supported by a grant from the Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number 15J06842).
Additional information
Funding
Notes on contributors
Naoya Yamaguchi
Naoya Yamaguchi was born in December 13, 1988 in Miyazaki, Japan. He obtained a degree in Physics at Kagoshima University, in March 2011. After he took a Master of Science in Mathematics at Kyushu University, in March 2014. Finally he obtained a Ph.D at Kyushu University, in March 2017. His research interests lie in the area of noncommutative determinant and group theory.
Notes
1. Here rings are assumed to possess a multiplicative unit.
2. The set and the product are explained in Section .
3. The algebra homomorphisms and are explained in Section .
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