<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$U_k(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>U</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the unitary associative algebra of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$k\times k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>×</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation> upper triangular matrices with entries from the field <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation> of complex numbers, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$F_k=F(U_k(\mathbb {C}))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mi>k</mi> </msub> <mo>=</mo> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the free algebra of rank 2 generated by the set <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\{u,v\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> in the variety generated by <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$U_k(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>U</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The algebra <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$F_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> satisfies the polynomial identity <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$[x_{1},x_{2}]\cdots [x_{2k-1},x_{2k}]=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo stretchy="false">]</mo> </mrow> <mo>⋯</mo> <mrow> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. The dihedral group <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$D_{2n}=\langle \rho ,\tau \mid \rho ^n=\tau ^2=(\tau \rho )^2=1\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo stretchy="false">⟨</mo> <mi>ρ</mi> <mo>,</mo> <mi>τ</mi> <mo>∣</mo> <msup> <mi>ρ</mi> <mi>n</mi> </msup> <mo>=</mo> <msup> <mi>τ</mi> <mn>2</mn> </msup> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mn>1</mn> <mo stretchy="false">⟩</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> acts on <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$F_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\rho (u)=e^{\frac{2\pi i}{n}}u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mrow> <mi>n</mi> </mfrac> </msup> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\rho (v)=e^{-{\frac{2\pi i}{n}}}v$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </msup> <mi>v</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\tau (u)=v$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>v</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\tau (v)=u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation>. In this study, we provide a generating set for the algebra <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$F_k^{D_{2n}}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>F</mi> <mi>k</mi> <msub> <mi>D</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </msubsup> </math></EquationSource> </InlineEquation> of invariants of <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$D_{2n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. We also compute the Hilbert series <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$H(F_k^{D_{2n}},t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <msubsup> <mi>F</mi> <mi>k</mi> <msub> <mi>D</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </msubsup> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$F_k^{D_{2n}}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>F</mi> <mi>k</mi> <msub> <mi>D</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </msubsup> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$k=3,4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Dihedral invariants in the variety of upper triangular matrices

期刊论文
发表日期：

2026年6月23日

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Şehmus Fındık

摘要

Let $U_k(\mathbb {C})$ $U_{k} (C)$ be the unitary associative algebra of $k\times k$ $k \times k$ upper triangular matrices with entries from the field $\mathbb {C}$ $C$ of complex numbers, and $F_k=F(U_k(\mathbb {C}))$ $F_{k} = F (U_{k} (C))$ be the free algebra of rank 2 generated by the set $\{u,v\}$ ${u, v}$ in the variety generated by $U_k(\mathbb {C})$ $U_{k} (C)$ . The algebra $F_k$ $F_{k}$ satisfies the polynomial identity $[x_{1},x_{2}]\cdots [x_{2k-1},x_{2k}]=0$ $[x_{1}, x_{2}] \dots [x_{2 k - 1}, x_{2 k}] = 0$ . The dihedral group $D_{2n}=\langle \rho ,\tau \mid \rho ^n=\tau ^2=(\tau \rho )^2=1\rangle $ $D_{2 n} = ⟨ ρ, τ ∣ ρ^{n} = τ^{2} = {(τ ρ)}^{2} = 1 ⟩$ acts on $F_k$ $F_{k}$ as $\rho (u)=e^{\frac{2\pi i}{n}}u$ $ρ (u) = e^{\frac{2 π i}{n}} u$ , $\rho (v)=e^{-{\frac{2\pi i}{n}}}v$ $ρ (v) = e^{- \frac{2 π i}{n}} v$ , $\tau (u)=v$ $τ (u) = v$ , $\tau (v)=u$ $τ (v) = u$ . In this study, we provide a generating set for the algebra $F_k^{D_{2n}}$ $F_{k}^{D_{2 n}}$ of invariants of $D_{2n}$ $D_{2 n}$ . We also compute the Hilbert series $H(F_k^{D_{2n}},t)$ $H (F_{k}^{D_{2 n}}, t)$ of $F_k^{D_{2n}}$ $F_{k}^{D_{2 n}}$ for $k=3,4$ $k = 3, 4$ .