fa توابع ماتریسی تعمیم یافته، دترمینان و پرمننت جبر alg علمی پژوهشی بنیادی S <span dir="RTL">در این مقاله با استفاده از ماتریس&shy;های جایگشتی یا ماتریس&shy;های متقارن شرایط لازم و کافی برای اینکه یک تابع ماتریسی تعمیم&shy; یافته دترمینان یا پرمننت باشد، ارائه می&shy;شود. ثابت می&shy;کنیم که یک تابع ماتریسی تعمیم&shy; یافته، دترمینان یا پرمننت است اگر و فقط اگر حافظ ضرب ماتریس&shy;های جایگشتی متقارن باشد. همچنین نشان می&shy;دهیم که یک تابع ماتریسی تعمیم &shy;یافته، دترمینان است اگر و فقط اگر حافظ ضرب ماتریس&shy;های متقارن باشد.</span> <span style="line-height:17.0pt"><b><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Introduction</span></span></b></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">Since linear and multilinear algebra has many applications in different branches of sciences, the attention of many mathematicians has been attracted to it in recent decades. The determinant and the permanent are the most important functions in linear algebra and so a generalized matrix function, which is a generalization of the determinant and the permanent, becomes significant. Generalized matrix functions connect some branches of mathematics such as theory of finite groups, representation theory of groups, graph theory and combinatorics, and linear and multilinear algebra. </span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">&nbsp;&nbsp; Let <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:20px; height:20px" ></span></span>&nbsp;be the symmetric group of degree <span style="position:relative"><span style="top:2.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image004.gif" style="width:15px; height:12px" ></span></span>, <span style="position:relative"><span style="top:2.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image006.gif" style="width:17px; height:16px" ></span></span>be a subgroup of <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:20px; height:20px" ></span></span>, and <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image009.gif" style="width:76px; height:20px" ></span></span>be a function. The function </span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image011.gif" style="width:117px; height:27px" ></span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">given by</span><span dir="RTL" lang="FA" style="font-size:10.0pt"><span style="font-family:"Arial",sans-serif"></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span style="position:relative"><span style="top:14.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image013.gif" style="width:178px; height:40px" ></span></span></span><span dir="RTL" lang="FA" style="font-size:10.0pt"><span style="font-family:"Arial",sans-serif"></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">is called the </span><span style="font-size:10.0pt">generalized matrix function associated to <span style="position:relative"><span style="top:2.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image006.gif" style="width:17px; height:16px" ></span></span>&nbsp;and </span><span style="font-size:10.0pt"><span style="font-family:Zar-s"><span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image016.gif" style="width:16px; height:16px" ></span></span></span></span><span style="font-size:10.0pt">. Note that if <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image018.gif" style="width:51px; height:20px" ></span></span>and <span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image020.gif" style="width:48px; height:22px" ></span></span>, the principal character of <span style="position:relative"><span style="top:2.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image006.gif" style="width:17px; height:16px" ></span></span>, then <span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image023.gif" style="width:65px; height:27px" ></span></span>is the permanent and&nbsp; if <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image018.gif" style="width:51px; height:20px" ></span></span>and <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image025.gif" style="width:43px; height:16px" ></span></span>, the alternating character of <span style="position:relative"><span style="top:2.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image006.gif" style="width:17px; height:16px" ></span></span>, then <span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image028.gif" style="width:64px; height:27px" ></span></span>is the determinant.</span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Results</span></b></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif">&nbsp;&nbsp;&nbsp; <span style="font-size:10.0pt">In this paper, using permutation matrices or symmetric matrices, necessary and sufficient conditions are given for a generalized matrix function to be the determinant or the permanent. We prove that a generalized matrix function is the determinant or the permanent if and only if it preserves the product of symmetric permutation matrices. Also we show that a generalized matrix function is the determinant if and only if it preserves the product of symmetric matrices. To be precise, we show that: </span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">If <b><i><span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image030.gif" style="width:51px; height:20px" ></span></span></i></b>and <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image009.gif" style="width:76px; height:20px" ></span></span>is a nonzero function, then the following are equivalent:</span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">1) </span><span style="font-size:10.0pt"><span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image028.gif" style="width:64px; height:27px" ></span></span>&nbsp;or <span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image023.gif" style="width:65px; height:27px" ></span></span>;</span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">2) </span><b><i><span style="font-size:10.0pt"><span style="position:relative"><span style="top:9.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image035.gif" style="width:266px; height:29px" ></span></span></span></i></b><span style="font-size:10.0pt">;</span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">3) </span><b><i><span style="font-size:10.0pt"><span euclid="" style="font-family:"><span style="position:relative"><span style="top:9.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image037.gif" style="width:268px; height:29px" ></span></span></span></span></i></b><span style="font-size:10.0pt">;</span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">where <b><i><span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image039.gif" style="width:22px; height:20px" ></span></span></i></b>is the permutation matrix induced by<b><i> <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image041.gif" style="width:48px; height:20px" ></span></span></i></b>and </span><b><i><span style="font-size:10.0pt"><span euclid="" style="font-family:"><span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image043.gif" style="width:152px; height:22px" ></span></span></span></span></i></b><span style="font-size:10.0pt">.</span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">Also if <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image030.gif" style="width:51px; height:20px" ></span></span>and <span style="position:relative"><span style="top:5.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image009.gif" style="width:76px; height:20px" ></span></span>is a nonzero function, then </span><span style="font-size:10.0pt"><span style="position:relative"><span style="top:7.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image028.gif" style="width:64px; height:27px" ></span></span>&nbsp;if and only if for all symmetric matrices </span><b><i><span style="font-size:10.0pt"><span euclid="" style="font-family:"><span style="position:relative"><span style="top:9.0pt"><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image045.gif" style="width:263px; height:29px" ></span></span></span></span></i></b><span style="font-size:10.0pt">.</span><span style="font-size:10.0pt"></span></span></span></span> تابع ماتریسی تعمیم یافته, دترمینان, پرمننت, ماتریس جایگشتی, ماتریس متقارن, کاراکتر تحویل ناپذیر, تابع کلاسی. Generalized matrix function, Determinant, Permanent, Permutation matrix, Symmetric matrix, Irreducible character, Class function. 76 88 http://mmr.khu.ac.ir/browse.php?a_code=A-10-1274-1&slc_lang=fa&sid=1 Mohammad Hossein Jafari محمد حسین جعفری jafari@tabrizu.ac.ir 10031947532846005463 10031947532846005463 Yes University of Tabriz دانشگاه تبریز Ali Reza Madadi علیرضا مددی a-madadi@tabrizu.ac.ir 10031947532846005464 10031947532846005464 No University of Tabriz دانشگاه تبریز