A non-abelian representation of the dual polar space DQ(2n,2)

We prove that the dual polar space $D Q (2 n, 2)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mi>Q</mi><mrow><mo class="MathClass-open">(</mo><mrow><mn>2</mn><mi>n</mi><mo class="MathClass-punc">,</mo><mn>2</mn></mrow><mo class="MathClass-close">)</mo></mrow></math>$ , $n \geq 3$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi> <mo class="MathClass-rel">≥</mo> <mn>3</mn></math>$ , of rank $n$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math>$ associated with a non-singular parabolic quadric in $PG (2 n, 2)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> PG</mo><mrow><mo class="MathClass-open">(</mo><mrow><mn>2</mn><mi>n</mi><mo class="MathClass-punc">,</mo><mn>2</mn></mrow><mo class="MathClass-close">)</mo></mrow></math>$ admits a faithful non-abelian representation in the extraspecial $2$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn></math>$ -group $2_{+}^{1 + 2^{n}}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mrow><mn>2</mn></mrow><mrow><mo class="MathClass-bin">+</mo></mrow><mrow><mn>1</mn><mo class="MathClass-bin">+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup> </mrow></msubsup></math>$ . The near $2 n$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mi>n</mi></math>$ -gon $I_{n}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi mathvariant="double-struck">I</mi></mrow><mrow><mi>n</mi></mrow></msub></math>$ (section 2.4) is a geometric hyperplane of $D Q (2 n, 2)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mi>Q</mi><mrow><mo class="MathClass-open">(</mo><mrow><mn>2</mn><mi>n</mi><mo class="MathClass-punc">,</mo><mn>2</mn></mrow><mo class="MathClass-close">)</mo></mrow></math>$ . For $n \geq 3$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi> <mo class="MathClass-rel">≥</mo> <mn>3</mn></math>$ , we first construct a faithful non-abelian representation of $I_{n}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi mathvariant="double-struck">I</mi></mrow><mrow><mi>n</mi></mrow></msub></math>$ in $2_{+}^{1 + 2^{n}}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mrow><mn>2</mn></mrow><mrow><mo class="MathClass-bin">+</mo></mrow><mrow><mn>1</mn><mo class="MathClass-bin">+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup> </mrow></msubsup></math>$ and subsequently extend it to a faithful non-abelian representation of $D Q (2 n, 2)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mi>Q</mi><mrow><mo class="MathClass-open">(</mo><mrow><mn>2</mn><mi>n</mi><mo class="MathClass-punc">,</mo><mn>2</mn></mrow><mo class="MathClass-close">)</mo></mrow></math>$ in $2_{+}^{1 + 2^{n}}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mrow><mn>2</mn></mrow><mrow><mo class="MathClass-bin">+</mo></mrow><mrow><mn>1</mn><mo class="MathClass-bin">+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup> </mrow></msubsup></math>$ .

Keywords

dual polar space, non-abelian representation, extraspecial 2-group

Mathematical Subject Classification 2000

Primary: 05B25

Milestones

Received: 20 July 2007

Accepted: 27 March 2008

Authors

Kamal Lochan Patra


Binod Kumar Sahoo

Vol. 9, No. 1, 2009

Kamal Lochan Patra and Binod Kumar Sahoo

Abstract

Keywords

Mathematical Subject Classification 2000

Milestones

Authors