# Questions tagged [quaternions]

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### Quotient rings of integral quaternion rings

I'm having a hard time finding information about the quotient rings of the Lipschitz quaternions and the Hurwitz quaternions. The Lipschitz quaternions are defined as the quaternions with integral ...
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1 vote
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### Is there a quaternionic analogue of Weyl's character formula?

I am pondering about this. The definition of a character makes sense for quaternionic matrices. Indeed, given a quaternionic representation of a quaternionic matrix group such as $GL(n, \mathbb{H})$ ...
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### Lifting of map from $S^3$ to itself

My question concerns the lifting of degree $0$ map from $S^3$ to itself. Let us suppose that all maps are smooth here. Looking at $S^3$ as the space of unit quaternions, one way to define degree is ...
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### Principled construction of the quaternions

Is there a construction of the quaternions that doesn't proceed through generators and relations, and which makes the connection with 3D rotations clear? I'm not happy with Clifford Algebra as an ...
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### Representing a number as a sum of four squares and factorization

Rabin and Shallit have a randomized polynomial-time algorithm to express an integer $n$ as a sum of four squares $n=a^2+b^2+c^2+d^2$ (in time $\log(n)^2$ assuming the Extended Riemann Hypothesis). I'...
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### Idempotent functions on Sp(1)

The quaternion group $Sp(1)\simeq S^3$ can be understood as $(z,w)\in\mathbb {C}^2$ with $|z|^2+|w|^2=1$ where multiplication is defined by $(z,w)(t,s)=(zt-\bar{s}w,zs+\bar{t}w)$. Question: How do ...
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### How many distinct quaternions have a given prime norm $p$?

I seem to recall that the answer is $p + 1$, but I'm not quite sure.
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### The name of special 16-dimensional hypercomplex number

Let's consider the following number: $n = n_0 + n_1\textbf{i} + n_2\textbf{j} + n_3\textbf{k}$ Where $\{1,\textbf{i},\textbf{j},\textbf{k}\}\subset\mathbb{H}$ and in turn each component can be written ...
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### Flag variety over quaternions and its Hecke algebra

Consider cell decomposition of flag variety of $\mathbb{H}^{n}$ into orbits under the action of $B(\mathbb{H})$ - group of upper triangular matrices with coefficients in $\mathbb{H}$. I think cells ...
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### The Hilbert symbols of quaternion algebras over a totally real field

Let $k$ be a totally real number field. Every quaternion algebra $B$ over $k$ can be written as $$B = \left(\frac{a,b}{k}\right),$$ for some constants $a,b \in k^\times$. My question is, can I always ...
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### Dimension of hermitian rank at most $k$ matrices over quaternions

In a (right) finite dimensional quaternionic Hilbert space there is an analogue of the spectral theorem (see theorem 4.6 in Farenick and Pidkowich) for normal matrices in $\mathbb{H}^{m\times m}$, ...
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### Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$
For a fixed positive integer $n$, the Diophantine equation $$x^2 + y^2 + z^2 = n$$ was studied by Gauss in Disquisitiones Arithmeticae. As is known, this equation is intimately connected to the ...