Study-type determinants and their properties

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.


Introduction
Let B be a commutative ring, let A be a ring 1  In the following, we assume that B is a free right C-module of rank n and C is a commutative ring. Then we have the following properties. 2 For all a ∈ M(r, A), the following hold: The Study-type determinant is a generalization of the Study determinant. The Study determinant was defined by Eduard Study [26]. Let H be the quaternion field. The Study determinant Sdet : M(r, H) → C is defined using a transformation from ψ r : M(r, H) → M(2r, C). It is known that this determinant has the following properties 3 (see, e.g., [2]).
For all a, a ′ ∈ M(r, H), the following hold: (S1 ′ ) Sdet(aa ′ ) = Sdet(a) Sdet(a ′ ); (S2 ′ ) a is invertible in M(r, H) if and only if Sdet(a) ∈ C is invertible; (S3 ′ ) if a ′ is obtained from a 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); (S4 ′ ) Sdet(a) ∈ Z(H) = R; (S5 ′ ) Det R M(4r,R) • φ 2r • ψ r (a) = Sdet(a) 2 . The above properties can be derived from the properties of the Study-type determinants.
Let L 2 and L 3 be left regular representations from B to M(n, C) and from M(r, B) to M(n, M(r, C)), respectively. The following theorem plays an important role in ascertaining the properties of the Study-type determinants. / / C Theorem 1.1 leads not only to some properties of the Study-type determinant, but also to Corollary 1.2 and Theorem 1.3. Let e = (e 1 e 2 · · · e m ) be a basis of A as B-module, let f = (f 1 f 2 · · · f m ) be a basis of B as C-module, let ef := {e i f j | i ∈ [m], j ∈ [n]}, let {x α | α ∈ ef } be the set of independent commuting variables, and let X ef = X := α∈ef αx α ∈ A[x α ] be the general element for ef , where A[x α ] is the polynomial ring in {x α | α ∈ ef } with coefficients in A. For rings R and R ′ , we denote the set of ring homomorphisms from R to R ′ by Hom(R, R ′ ), and we regard any ring homomorphism ρ ∈ Hom(R, R ′ ) as ρ ∈ Hom(R[x α ], R ′ [x α ]) such that ρ(x α 1 R ) = x α ρ(1 R ) for any α ∈ ef , where 1 R is the unit element of R. Let ρ ∈ Hom(A, M(r, B)). We assume that there exists a commutative ring such that C ⊂C, and L 3 • ρ and L 2 have the direct sums where a ∈ A and b ∈ B. Then, we have the following corollary and theorem.
M(r,B[xα]) • ρ (X). Corollary 1.2 leads to an extension of Dedekind's theorem, while Theorem 1.3 leads to an inequality characterizing the degrees of irreducible representations of finite groups. Let Θ(G) be the group determinant of the finite group G, let G be a complete set of inequivalent irreducible representations of G over C, let {x g | g ∈ G} be independent commuting variables, let C[x g ] = C [x g ; g ∈ G] be the polynomial ring in {x g | g ∈ G} with coefficients in C, let CG be the group algebra of G over C, let C[x g ]G := C[x g ] ⊗ CG = g∈G c g g | c g ∈ C[x g ] , let 1 G be the unit element of G, and let |G| be the order of G. We extend ϕ ∈ G to ϕ : The extension of Dedekind's theorem mentioned above is the following.
The group determinant Θ(G) ∈ C[x g ] is the determinant of a matrix with entries in {x g | g ∈ G}. (It is known that the group determinant determines the group. For the details, see [8] and [23].) Dedekind proved the following theorem concerning the irreducible factorization of the group determinant for any finite abelian group (see, e.g., [28]). Theorem 1.5 (Dedekind's theorem). Let G 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., [5]  Note that Corollary 1.8 follows from Frobenius reciprocity, and it is known that if H is an abelian normal subgroup of G, then deg ϕ divides [G : H] for all ϕ ∈ G (see e.g, [18]). This paper is organized as follows. In Section 2, 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 1.1. In Section 3, 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 1.1. In Section 4, we prove Theorem 1.1. In Section 5, we give a corollary concerning the degrees of some representations contained in regular representations. In Section 6, 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 7, we give a Cayley-Hamilton-type theorem for the Study-type determinant under the assumption that there exists a basis of A as a B-right module satisfying the conditions (i) and (ii). This Cayley-Hamilton type theorem leads to some properties of Study-type determinants. In Section 8, we obtain two expressions for regular representations under the assumption that the basis e = (e 1 e 2 · · · e m ) of A as a B-module satisfying the following conditions: In addition, we characterize the images of regular representations in the case that e satisfies the following additional condition: (vi) for any e i and e j , e i B * e j B = e j B * e i B.
This characterization is the following. In Section 9, 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 1.9, we obtain the following characterizations of φ r and ψ r .
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.

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 4.4.

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 R be a ring, which is assumed to have a multiplicative unit 1, let M(m, R) be the set of all m × m matrices with elements in R, let X = (X ij ) 1≤i,j≤m ∈ M(m, R), let [m] := {1, 2, . . . , m}, and let S m be the symmetric group on [m]. We express the determinant of X from M(m, R) to R as The group S m acts on M(m, R) as σ · X := (X σ(i)σ(j) ) 1≤i,j≤m , where σ ∈ S m . If R is commutative, then the group action does not change the determinants of matrices in M(m, R). 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 X = (X ij ) 1≤i,j≤mn ∈ M(mn, R). The mn × mn matrix X can be written as 1≤k,l≤n , where X (k,l) are m × m matrices. The following is a known theorem concerning commuting-block matrices [15] and [19]. Theorem 2.1. Let R be a commutative ring, and assume that X (k,l) ∈ M(m, R) are commutative. Then we have Let I k be the identity matrix of size k. We have the following lemma.
Lemma 2.2. Let R ′ be a ring, let R be a commutative ring, let S be a subring of R, and let η be a ring homomorphism from Proof. (The method used here is based on that of the proof of Theorem 2.1 in [19].) We prove this by induction on n. In the case n = 1, the statement is obviously true. Then, assuming that the statement is true for n − 1, we prove it for n.
Then we find that the following equation holds: , and replace X 11 by x + X 11 . Then, because Det R M(m,R) (η(x + X 11 )) is neither zero nor a zero divisor, we have . Substituting x = 0 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 σ(m, n) ∈ S mn using the Euclidean algorithm. This permutation reverses the order of the Kronecker product.
Next, we determine a permutation using the Euclidean algorithm. From the Euclidean algorithm, we know that σ(m, n) : is a bijection map, where k ∈ [n] and l ∈ [m]. Thus, σ(m, n) ∈ S mn . We have the following lemma.
Proof. For any p, q ∈ [mn], by the Euclidean algorithm there exist unique integers k, l ≥ 1 and s, t ∈ [m] such that p = m(k − 1) + s and q = m(l − 1) + t. 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., [14]). We do not explain this general property, because, for our purposes, it is simpler to use Lemma 2.3.

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 A and B be rings and let Z(A) be the center of A. Assume that A is a free right B-module with an ordered basis e = (e 1 e 2 · · · e m ). In other words, A = i∈[m] e i B, and B is a subring of A. Then, for all a ∈ A, there exists a unique (b ij ) 1≤i,j≤m ∈ M(m, B) such that Let R be the field of real numbers, let C be the field of complex numbers, and let H := {1a + ib + jc + kd | a, b, c, d ∈ R} be the quaternion field. Below, we give three examples of regular representations.
where b is the complex conjugate matrix of b ∈ B.

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., [3]). However, we do not assume that invertibility preserving maps are linear maps as in [31].
The following is the definition of invertibility preserving maps.
Definition 3.4 (Invertibility preserving map). Let R and R ′ be rings, and let η : R → R ′ be a map. Assume that for any α ∈ R, the following condition holds: α is invertible in R if and only if η(α) is invertible in R ′ . Then we call η an "invertibility preserving map." We recall that if B is a commutative ring, then we do not need to distinguish between left and right inverses for a ∈ A. Because, L e is an injective algebra homomorphism, and if L e (a)L e (b) is the unit element, then L e (b)L e (a) is the unit element.
We denote the unit element of A as 1. In terms of the regular representation, we have the following lemma.  Proof. For all a ∈ A, we have This completes the proof.
Let r ∈ N and let L e⊗Ir be the left regular representation from M(r, A) to M(m, M(r, B)) with respect to e ⊗ I r . Then, from Lemma 3.6, we have the following corollary.

A commutative diagram on the regular representations and determinants
In this section, we prove Theorem 4.4. 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 6. In addition, from this commutative diagram, we are able to derive an inequality for the degrees of representations (Section 5) and an extension of Dedekind's theorem (Section 10).
Let E ij be the r × r matrix with 1 in the (i, j) entry and 0 otherwise. First, we prove the following lemma: L e (a ij ) ⊗ E ij .
Proof. We express (L e (a ij ) kl ) 1≤i,j≤r as a (e k L e (a ij ) kl ) 1≤i,j≤r = (a ij e l ) 1≤i,j≤r = ae l I r .
Therefore, we have (e ⊗ I r ) a  E ij ⊗ L e (a ij ) = (L e (a ij )) 1≤i,j≤r .
Next, from Lemmas 2.2 and 4.2, we obtain the following corollary.
This completes the proof.

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.
, j ∈ [n]} and let [x α ] := {x α | α ∈ ef } be the set of independent commuting variables, and we denote X ef ∈ (ef )[x α ] ⊂ A[x α ] as X. For rings R and R ′ , we denote the set of ring homomorphisms from R to R ′ by Hom(R, R ′ ), and we regard any ring homomorphism ρ ∈ Hom(R, From Theorem 4.4, for any ρ ∈ Hom(A, M(r, B)), we have M(r,B[xα]) • ρ (X). LetC be a commutative ring such that C ⊂C. We assume that L f ⊗Ir • ρ and L f have the following direct sums: where a ∈ A and b ∈ B. Then we have the following corollary from Theorem 4.4.

Proof. If DetC
This completes the proof.

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. 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. For a ring R and k ∈ N, we denote M(k, R) as M k (R). From Corollary 3.7 and Theorem 4.4, we obtain the following commutative diagram:

Mmnr (C)
/ / C From this, we obtain the following theorem.

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 A as B-module satisfying certain conditions. This Cayley-Hamilton-type theorem leads to some properties of Study-type determinants.
Let L be a left regular representation from A to M(m, B) and let x be an independent variable. We write (Det B A • ι)(xI m − L(a)) ∈ B[x] as Φ L(a) (x) for all a ∈ A, i.e., we express the characteristic polynomial of L(a) as Φ L(a) (x). In this section, we assume that e has the following properties: i Be i ⊂ B holds for any e i , Then we have the following lemma.
Proof. Without loss of generality, we can assume that L = L e . We show that e −1 i Φ L(a) (x)e i = Φ L(a) (x) for all e i . Since e i is invertible for all e i , there exists an invertible element Q i ∈ M(m, B) such that ee i = eQ i . Then, from ae = ee i e −1 i L(a)e i e −1 i , we obtain This completes the proof.
The following theorem is a Cayley-Hamilton-type theorem. Proof. Without loss of generality, we can assume that L = L e . From the Cayley-Hamilton theorem for commutative rings, we have L(a) m + b m−1 L(a) m−1 + · · · + b 0 I m = 0. Then, because L is a Z(A) ∩ B-algebra homomorphism, from Lemma 7.1, we obtain Finally, because L is injective, we have Φ L(a) (a) = 0. This completes the proof.
From Corollary 4.3 and Lemma 7.1, we obtain the following corollary.

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 A as B-module satisfies certain conditions. Let eB := {e i B | i ∈ [m]} and we define a product of e i B and e j B as In this section, we assume that e satisfies the following conditions: (iii) for any e i and e j , e i B * e j B ∈ eB; (iv) there exists e k such that e k B = B as set; (v) for any e i , there exists e j such that e i B * e j B = B. It is easy to show that e i is invertible in A and (eB, * ) is a group. For a group G, we denote the unit element of G as 1 G . We remark that even if e ′ has the above properties, then eB and e ′ B are not necessary group isomorphism. Let P (e) be the diagonal matrix diag(e 1 , e 2 , . . . , e m ) ∈ M(m, A). To obtain an expression for L e , we define the indicator function 1 B by We now formulate an expression for regular representations in terms of 1 B .
This implies b = L e (a) ∈ L e (A). This completes the proof.
From Lemma 4.1 and Theorem 8.4, we obtain the following corollary.

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 r × r matrix can be written uniquely as b = c 1 + ic 2 , where c 1 and c 2 ∈ M(r, R), and any quaternionic r × r matrix can be written uniquely as a = b 1 + jb 2 , where b 1 and b 2 ∈ M(r, C). We define φ r : M(r, C) → M(2r, R) and ψ r : M(r, H) → M(2r, C) by Then the following are known (see, e.g., [2]): (S0 ′ ) the maps φ r and ψ r are injective algebra homomorphisms; (S1 ′ ) Sdet(aa ′ ) = Sdet(a) Sdet(a ′ ); (S2 ′ ) a is invertible in M(r, H) if and only if Sdet(a) ∈ C is invertible; (S3 ′ ) if a ′ is obtained from a 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);

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 G be a finite group, let [x g ] = {x g | g ∈ G} be the set of independent commuting variables, let C[x g ] be the polynomial ring in the variables [x g ] with coefficients in C, and let |G| be the order of G. The group determinant Θ(G) of G is given by Θ(G) := Det where we apply a numbering to the elements of G (for details, see, e.g., [5], [9], [10], [11], [13], [17], [28], and [29]). It is thus seen that the group determinant Θ(G) is a homogeneous polynomial of degree |G|. In general, the matrix (x gh −1 ) g,h∈G is covariant under a change in the numbering of the elements of G. However, the group determinant, Θ(G), is invariant.
Let CG := g∈G c g g | c g ∈ C be the group algebra of G over C, let H be an abelian subgroup of G, let [G : H] be the index of H in G, let A = C[x g ] ⊗ CG = g∈G c g g | c g ∈ C[x g ] , let B = C[x g ] ⊗ CH, and let C = C[x g ] ⊗ C{1 G }. For a group G, we denote the regular representation of the group G as L G . We regard L G as C[x g ]algebra homomorphism from A to C. Then, from Lemma 8.2, L G is equivalent to the left regular representation from A to M(|G|, C). Therefore, the following commutative diagram holds: g∈G x g L G (g) (see, e.g., [5]). Therefore We extend ϕ ∈ G to ϕ : C[x g ]G → C[x g ]G satisfy ϕ g∈G c g g = g∈G c g ϕ(g), where c g ∈ C[x g ]. Frobenius proved the following theorem concerning the factorization of the group determinant (see, e.g., [5]). Let G = {ϕ (1) , ϕ (2) , . . . , ϕ (s) } be a complete set of inequivalent irreducible representations of G over C. Theorem 10.1 holds from the following theorem (which is treated in detail in [25]).
Therefore, the following theorem is deduced from Corollary 5.2. The following is a special case of Theorem 10.3 [29]. Note that Corollary 10.5 follows from Frobenius reciprocity, and it is known that if H is an abelian normal subgroup of G, then deg ϕ divides [G : H] for all ϕ ∈ G (see, e.g, [18]).