As a former recruiter, I have a few thoughts:
-If you aren't interested in jobs in that division, you should ask the recruiter to withdraw your resume. Recruiters often don't have all the details about positions (for various reasons: the company doesn't provide much, they are working through a middle man, the recruiter is sketchy, or anything in between) and will submit candidates for positions that aren't perfect fits. Recruiting/staffing (not corporate recruiters) is a numbers game--they get judged on resume submittals, interviews, and ultimately job placements of their candidates. What recruiters want to avoid is having a candidate get an interview request, and the candidate turn down the request. Make the recruiter withdraw your resume if you want to avoid that situation.
- If you are interested in the job: how close are you with the Senior Executive contact? If it is someone who you are comfortable reaching out to, you should absolutely do it. A personal connection will go miles towards getting an interview.
-Does the Senior Executive know you are interested in jobs at the company? If so, do they know about your background and if you're qualified? It's not a great sign that the company is still using outside staffing firms to recruit for the position.
-At the end of the day, that Senior Executive can do much more towards getting you an interview (note that I said interview, not job) than any recruiter can. You need to cultivate that relationship if it is viable.
Companies that use outside recruiters often have many recruiting firms working on their openings, and recruiters will indiscriminately send in resumes to see if they get any traction.
Good luck!