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| {{group-specific information|
| | #redirect [[Linear representation theory of modular maximal-cyclic group:M16]] |
| information type = linear representation theory|
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| group = M16|
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| connective = of}}
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| This article discusses the linear representation theory of the group [[M16]], a group of order 16 given by the [[presentation]]:
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| <math>G = M_{16} := \langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^5 \rangle</math>
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| ==Summary==
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| {| class="sortable" border="1"
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| ! Item !! Value
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| | [[degrees of irreducible representations]] over a [[splitting field]] (such as <math>\overline{\mathbb{Q}}</math> or <math>\mathbb{C}</math>) || 1,1,1,1,1,1,1,1,2,2 (1 occurs 8 times, 2 occurs 2 times)<br>[[maximum degree of irreducible representation|maximum]]: 2, [[lcm of degrees of irreducible representations|lcm]]: 2, [[number of irreducible representations equals number of conjugacy classes|number]]: 10, [[sum of squares of degrees of irreducible representations equals order of group|sum of squares]]: 16
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| | [[Schur index]] values of irreducible representations || 1 (all of them)
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| | smallest ring of realization (characteristic zero) || <math>\mathbb{Z}[i] = \mathbb{Z}[\sqrt{-1}] = \mathbb{Z}[t]/(t^2 + 1)</math> -- [[ring of Gaussian integers]]
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| | [[ring generated by character values]] (characteristic zero) || <math>\mathbb{Z}[2i] = \mathbb{Z}[\sqrt{-4}] = \mathbb{Z}[t]/(t^2 + 4)</math>
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| | [[minimal splitting field]], i.e., smallest field of realization (characteristic zero) || <math>\mathbb{Q}(i) = \mathbb{Q}(\sqrt{-1}) = \mathbb{Q}[t]/(t^2 + 1)</math><br>Same as [[field generated by character values]], because all Schur index values are 1.
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| | condition for a field to be a [[splitting field]] || The characteristic should not be equal to 2, and the polynomial <math>t^2 + 1</math> should split.<br>For a finite field of size <math>q</math>, this is equivalent to saying that <math>q \equiv 1 \pmod 4</math>
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| | [[minimal splitting field]] in characteristic <math>p \ne 0,2</math> || Case <math>p \equiv 1 \pmod 4</math>: prime field <math>\mathbb{F}_p</math><br>Case <math>p \equiv 3 \pmod 4</math>: Field <math>\mathbb{F}_{p^2}</math>, quadratic extension of prime field
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| | smallest size splitting field || [[Field:F5]], i.e., the field with five elements.
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| | degrees of irreducible representations over the rational numbers || 1,1,1,1,2,2,4 (1 occurs 4 times, 2 occurs 2 times, 4 occurs 1 time)
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| |}
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| ==Representations==
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| ===Summary information===
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| {| class="sortable" border="1"
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| ! Name of representation type !! Number of representations of this type !! Values not allowed for field characteristic !! Criterion for field !! What happens over a splitting field? !! Kernel !! Quotient by kernel (on which it descends to a faithful representation) !! [[Degree of a linear representation|Degree]] !! [[Schur index]] !! What happens by reducing the <math>\mathbb{Z}</math> or <math>\mathbb{Z}[i]</math>-representation over bad characteristics?
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| | trivial || 1 || -- || any || remains the same || whole group || [[trivial group]] || 1 || 1 || no bad characteristics
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| | sign, kernel a cyclic maximal subgroup || 2 || -- || any || remains the same || [[Z8 in M16]] -- either <math>\langle a \rangle</math> or <math>\langle ax \rangle</math> || [[cyclic group:Z2]] || 1 || 1 || no bad characteristics, but reduced mod 2, this representation becomes the trivial representation
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| | sign, kernel a non-cyclic maximal subgroup || 1 || -- || any || remains the same || [[direct product of Z4 and Z2 in M16]] -- <math>\langle a^2, x \rangle</math> || [[cyclic group:Z2]] || 1 || 1 || no bad characteristics, but reduced mod 2, this representation becomes the trivial representation
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| | representation with kernel <math>\langle a^2x \rangle</math> || 2 || -- || must contain a primitive fourth root of unity, or equivalently, <math>t^2 + 1</math> must split || remains the same || [[non-central Z4 in M16]] || [[cyclic group:Z4]] || 1 || 1 || no bad characteristics, but in characteristic 2, becomes the same as the trivial representation.
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| | representation with kernel <math>\langle a^2, x \rangle</math> || 2 || -- || must contain a primitive fourth root of unity, or equivalently, <math>t^2 + 1</math> must split || remains the same || [[V4 in M16]] || [[cyclic group:Z4]] || 1 || 1 || no bad characteristics, but in characteristic 2, becomes the same as the trivial representation.
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| | [[faithful irreducible representation of M16]] || 2 || 2 || must contain a primitive fourth root of unity, or equivalently, <math>t^2 + 1</math> must split || remains the same || trivial subgroup || [[M16]] || 2 || 1 || in characteristic 2, acquires a kernel of order eight, and we get an indecomposable but not irreducible representation of the quotient group.
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| | ''not over a splitting field'': four-dimensional representation || 1 || 2 || must ''not'' contain a primitive fourth root of unity, or equivalently, <math>t^2 + 1</math> does not split || splits into the two faithful irreducible representations of degree two || trivial subgroup || [[M16]] || 4 || 1 ||
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| |}
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| ==Character table==
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| {{character table facts to check against}}
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| Below is the character table over a [[splitting field]]. Here <math>i</math> denotes a square root of <matH>-1</math> in the field.
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| {| class="sortable" border="1"
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| ! Representation/conjugacy class and size !! <math>\{ e \}</math> (size 1) !! <math>\{ a^4 \}</math> (size 1) !! <math>\{ a^2 \}</math> (size 1) !! <math>\{ a^6 \}</math> (size 1) !! <math>\{ a, a^5 \}</math> (size 2) !! <math>\{ a^3, a^7 \}</math> (size 2) !! <math>\{ ax, a^5x \}</math> (size 2) !! <math>\{ a^3x, a^7x \}</math> (size 2) !! <math>\{ x, a^4x\}</math> (size 2) !! <math>\{ a^2x, a^6x \}</math> (size 2)
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| | trivial || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
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| | <math>\langle a \rangle</math>-kernel || 1 || 1 || 1 || 1 || 1 || 1 || -1 || -1 || -1 || -1
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| | <math>\langle ax \rangle</math>-kernel || 1 || 1 || 1 || 1 || -1 || -1 || 1 || 1 || -1 || -1
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| | <math>\langle a^2,x \rangle</math>-kernel || 1 || 1 || 1 || 1 || -1 || -1 || -1 || -1 || 1 || 1
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| | <math>\langle a^2x \rangle</math>-kernel (first) || 1 || 1 || -1 || -1 || <math>i</math> || <math>-i</math> || <math>-i</math> || <math>i</math> || -1 || 1
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| | <math>\langle a^2x \rangle</math>-kernel (second) || 1 || 1 || -1 || -1 || <math>-i</math> || <math>i</math> || <math>i</math> || <matH>-i</math> || -1 || 1
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| | <math>\langle a^4,x \rangle</math>-kernel (first) || 1 || 1 || -1 || -1 || <math>i</math> || <math>-i</math> || <math>i</math> || <math>-i</math> || 1 || -1
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| | <math>\langle a^4,x \rangle</math>-kernel (second) || 1 || 1 || -1 || -1 || <math>-i</math> || <math>i</math> || <math>-i</math> || <math>i</math> || 1 || -1
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| | [[faithful irreducible representation of M16]] (first)|| 2 || -2 || <math>2i</math> || <math>-2i</math> || 0 || 0 || 0 || 0 || 0 || 0
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| | [[faithful irreducible representation of M16]] (second) || 2 || -2 || <math>-2i</math> || <math>2i</math> || 0 || 0 || 0 || 0 || 0 || 0
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| |}
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| Below are the size-degree-weighted characters, obtained by multiplying each character value by the size of the conjugacy class and dividing by the degree of the irreducible representation:
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| {| class="sortable" border="1"
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| ! Representation/conjugacy class and size !! <math>\{ e \}</math> (size 1) !! <math>\{ a^4 \}</math> (size 1) !! <math>\{ a^2 \}</math> (size 1) !! <math>\{ a^6 \}</math> (size 1) !! <math>\{ a, a^5 \}</math> (size 2) !! <math>\{ a^3, a^7 \}</math> (size 2) !! <math>\{ ax, a^5x \}</math> (size 2) !! <math>\{ a^3x, a^7x \}</math> (size 2) !! <math>\{ x, a^4x\}</math> (size 2) !! <math>\{ a^2x, a^6x \}</math> (size 2)
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| | trivial || 1 || 1 || 1 || 1 || 2 || 2 || 2 || 2 || 2 || 2
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| | <math>\langle a \rangle</math>-kernel || 1 || 1 || 1 || 1 || 2 || 2 || -2 || -2 || -2 || -2
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| | <math>\langle ax \rangle</math>-kernel || 1 || 1 || 1 || 1 || -2 || -2 || 2 || 2 || -2 || -2
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| | <math>\langle a^2,x \rangle</math>-kernel || 1 || 1 || 1 || 1 || -2 || -2 || -2 || -2 || 2 || 2
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| | <math>\langle a^2x \rangle</math>-kernel (first) || 1 || 1 || -1 || -1 || <math>2i</math> || <math>-2i</math> || <math>-2i</math> || <math>2i</math> || -2 || 2
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| | <math>\langle a^2x \rangle</math>-kernel (second) || 1 || 1 || -1 || -1 || <math>-2i</math> || <math>2i</math> || <math>2i</math> || <math>-2i</math> || -2 || 2
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| | <math>\langle a^4,x \rangle</math>-kernel (first) || 1 || 1 || -1 || -1 || <math>2i</math> || <math>-2i</math> || <math>2i</math> || <math>-2i</math> || 2 || -2
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| | <math>\langle a^4,x \rangle</math>-kernel (second) || 1 || 1 || -1 || -1 || <math>-2i</math> || <math>2i</math> || <math>-2i</math> || <math>2i</math> || 2 || -2
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| | [[faithful irreducible representation of M16]] || 1 || -1 || <math>i</math> || <math>-i</math> || 0 || 0 || 0 || 0 || 0 || 0
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| | [[faithful irreducible representation of M16]] (second) || 1 || -1 || <math>-i</math> || <math>i</math> || 0 || 0 || 0 || 0 || 0 || 0
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| |}
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